Islam places an enormous amount of respect to the parents.  To the father and the mother.  In a lecture giving by Sheikh Amini, congregational imam of Imam Mahdi School in Qom, he describes the perfection that is to be reached by serving you parents, and in the case of the story in question, your mother.  The audio is in farsi,  with English transcript below:

Title: Seek Perfection Through Devotion to Your Mother

00:35 One of the greats,
00:38 perhaps you were there for one of the trips we went on to Mashad, I don’t know,
00:42 It was in Sabzevar, if I am not mistaken, or Shahrud
00:49 We went there, there was one school, which taught for over 700 years, a scholarly school , built
00:57 over 700 years ago, and students there would study, and still do 1.03 we said why would you study here this building could collapse!, that’s how old it was.
1.06 they said, no Sir, don’t stress about that
1.12 One of the Awaliya Allah was buried at the school, Bayazid Bastami, known as Bayazid Bastami
1.22 Who was he? Someone who would say when I was with my mother and went for days on end to study with my teacher, who taught me from the Quran Surah Hamd
1.35 Taught me and taught me and taught me until we reached Suarh Anaam
1.41 قُلِ اللَّـهُ ۖ ثُمَّ ذَرْهُمْ – Say: ‘God.’ Then leave them alone (Arberry Translation), one of the Ayahs somewhere in the middle of Surah Alanaam
1.46 Say God and anything other than God let go off.
1.52 The teacher then said to me, at this point, either God, or others.
1.58 So I went to my mother and said
2.01 My dear mother, should I want to come to you everyday, I’ll fall behind in my studies
2.06 would you grant me permission, for example, to not come to you?
2.09 This is the mother here! A mother can’t just let go of her children.
2.12 but she saw that her child made the request and
2.15 She said – go my dear, go where ever you wish to go and study, God willing, you will obtain high levels.
2.20 So Bayazid Bastami says, I left my oppressed mother, imagining that
2.28 I would go and study
2.32 Thirty years, moving from this school to that school and from one
town to another town 2.38 Until he said, when thirty years have passed, he intended to perform pilgrimage and after that
2.44 return to his town, to attend to my mother
2.48 on a snowy winters night I arrived at the door of the house, I saw that my mother was awake
2.55 and was supplicating to God… (to) please bring home this traveller (ghareeb) of mine
This traveller of mine – her child
3.06 I knocked on the door, “who is it?” she said.
3.08 I replied, “My mother, your traveller (ghareeb) has returned”
3.10 happily she opened the door and…
3.14 when the mother was asleep errr….
3.17 (she) woke up (and she said to her son) – “I am thirsty”.
3.19 There wasn’t water in the house, (so he) went outside the house (to fetch water), the pool had frozen over
3.25 he broke through the ice and brought the water in, but saw that his mother had fallen back asleep.
3.30 He held the water in his hand next to his mother, awaiting his mother to wake up
3.35 once his mother woke up, he gave her the water, and she supplicated for him
3.40 he said when I served my mother, what I learnt, and the perfection, the for thirty years I was chasing that spiritual perfection,
3.52 in the service of the mother, the perfection was granted to me!!!
4.08 you see, my dear friends, study they do, (audience are students of Islamic Seminaries)
4.12 if that which is their duty, they do not fulfil.
4.14 Duty
4.16 That which will bring us closer to God, is fulfillment of the duty (we have)
4.21 Yes, in any case, studying for some is a duty, but if this duty
4.27 alongside studying, there is another duty, that too must be taken care of.
4.31 That is how one, with research, intellectual though, and seeking advice, God willing, can reach levels of nearness to The High God.
4.40 My Lord! This educator and tutor of ours, Imam Sajjad Peace be Upon Him,
4.42 Supplicating to the Lord
وَاَنَّ الِراحِلَ اِلَيْكَ قَريبُ الْمَسافَةِ
The one that moves towards you will reach their destination quickly
4.56 For 30 years, he was going in another direction.
وَاَنَّكَ لا تَحْتَجِبُ عَنْ خَلْقِكَ
إلاّ اَنْ تَحْجُبَهُمُ الأعمالُ دُونَكَ
There is no veil or barrier between you and your creation
Except for their impure deeds
5.15 God willing, may our Lord grace us all with knowledge and deeds
(based on that knowledge) 5.19 with the blessings upon the Prophet Mohammad and his Purified Household.

It is said that Imam Sadiq, peace be upon him, answered the question as to what is the best of deeds with the following:

 قال: الصلاة لوقتها، وبر الوالدين، والجهاد في سبيل الله
Prayers on time, devotion to both parents, and Jihad in the path of Allahﷻ.

In another narration, the Prophet, blessings be upon him and his Purified Household, is recorded to have said:

وقال رسول الله (ص): بر الوالدين أفضل من الصلاة والصوم والحج والعمرة والجهاد في سبيل الله 
Devotion to the two parents (father and mother) is better than prayers, and fasting, and pilgrimage, and the Ummra, and Jihad in the path of Allahﷻ.

The above narrations are from (جامع السعادات).  

The Holy Qur’an has, in numerous places, mentioned the need to be good to parents right after the command of monotheism.  Such as in the following Ayahs:

1) وَقَضَىٰ رَبُّكَ أَلَّا تَعْبُدُوا إِلَّا إِيَّاهُ وَبِالْوَالِدَيْنِ إِحْسَانًا إِمَّا يَبْلُغَنَّ عِندَكَ الْكِبَرَ أَحَدُهُمَا أَوْ كِلَاهُمَا فَلَا تَقُل لَّهُمَا أُفٍّ وَلَا تَنْهَرْهُمَا وَقُل لَّهُمَا قَوْلًا كَرِيمًا﴿الإسراء: ٢٣﴾
Thy Lord has decreed you shall not serve any but Him, and to be good to parents, whether one or both of them attains old age with thee; say not to them ‘Fie’ neither chide them, but speak unto them words respectful, (17: 23)
(2) وَإِذْ أَخَذْنَا مِيثَاقَ بَنِي إِسْرَائِيلَ لَا تَعْبُدُونَ إِلَّا اللَّهَ وَبِالْوَالِدَيْنِ إِحْسَانًا وَذِي الْقُرْبَىٰ وَالْيَتَامَىٰ وَالْمَسَاكِينِ وَقُولُوا لِلنَّاسِ حُسْنًا وَأَقِيمُوا الصَّلَاةَ وَآتُوا الزَّكَاةَ ثُمَّ تَوَلَّيْتُمْ إِلَّا قَلِيلًا مِّنكُمْ وَأَنتُم مُّعْرِضُونَ﴿البقرة: ٨٣﴾
And when We took compact with the Children of Israel: ‘You shall not serve any save God; and to be good to parents, and the near kinsman, and to orphans, and to the needy; and speak good to men, and perform the prayer, and pay the alms.’ Then you turned away, all but a few of you, swerving aside. (2: 83)
(3) وَاعْبُدُوا اللَّهَ وَلَا تُشْرِكُوا بِهِ شَيْئًا وَبِالْوَالِدَيْنِ إِحْسَانًا وَبِذِي الْقُرْبَىٰ وَالْيَتَامَىٰ وَالْمَسَاكِينِ وَالْجَارِ ذِي الْقُرْبَىٰ وَالْجَارِ الْجُنُبِ وَالصَّاحِبِ بِالْجَنبِ وَابْنِ السَّبِيلِ وَمَا مَلَكَتْ أَيْمَانُكُمْ إِنَّ اللَّهَ لَا يُحِبُّ مَن كَانَ مُخْتَالًا فَخُورًا﴿النساء: ٣٦﴾ 
Serve God, and associate naught with Him. Be kind to parents, and the near kinsman, and to orphans, and to the needy, and to the neighbour who is of kin, and to the neighbour who is a stranger, and to the companion at your side, and to the traveller, and to that your right hands own. Surely God loves not the proud and boastful (4: 36)
(4) قُلْ تَعَالَوْا أَتْلُ مَا حَرَّمَ رَبُّكُمْ عَلَيْكُمْ أَلَّا تُشْرِكُوا بِهِ شَيْئًا وَبِالْوَالِدَيْنِ إِحْسَانًا وَلَا تَقْتُلُوا أَوْلَادَكُم مِّنْ إِمْلَاقٍ نَّحْنُ نَرْزُقُكُمْ وَإِيَّاهُمْ وَلَا تَقْرَبُوا الْفَوَاحِشَ مَا ظَهَرَ مِنْهَا وَمَا بَطَنَ وَلَا تَقْتُلُوا النَّفْسَ الَّتِي حَرَّمَ اللَّهُ إِلَّا بِالْحَقِّ ذَٰلِكُمْ وَصَّاكُم بِهِ لَعَلَّكُمْ تَعْقِلُونَ﴿الأنعام: ١٥١﴾ 
Say: ‘Come, I will recite what your Lord has forbidden you: that you associate not anything with Him, and to be good to your parents, and not to slay your children because of poverty; We will provide you and them and that you approach not any indecency outward or inward, and that you slay not the soul God has forbidden, except by right. That then He has charged you with; haply you will understand. (6: 151)
Practical Application:  There is no doubt, and the narrations are plenty, that we need to be devoted to the service of our parents.  Even if they are disbelievers, our parents are sanctified in Islam and are given an esteem so high, it comes next to serving Allahﷻ in the Holy Qur’an, and is deemed to be better than your prayers or fasts or other worships.