At the outset, it is important to understand that Allah (God) has ordered to us to respect both parents at all times.
The Almighty says in the Holy Quran:
وَقَضَى رَبُّكَ أَلَّا تَعْبُدُوا إِلَّا إِيَّاهُ وَبِالْوَالِدَيْنِ إِحْسَانًا إِمَّا يَبْلُغَنَّ عِنْدَكَ الْكِبَرَ أَحَدُهُمَا أَوْ كِلَاهُمَا فَلَا تَقُلْ لَهُمَا أُفٍّ وَلَا تَنْهَرْهُمَا وَقُلْ لَهُمَا قَوْلًا كَرِيمًا (23) وَاخْفِضْ لَهُمَا جَنَاحَ الذُّلِّ مِنَ الرَّحْمَةِ وَقُلْ رَبِّ ارْحَمْهُمَا كَمَا رَبَّيَانِي صَغِيرًا
And your Lord has decreed that you worship none but Him and do good to parents. If either of them or both reach old age, do not say to them 'uff' (a word of anger or contempt) and do not scold them. And address them with respectful words, and submit yourself before them in humility out of compassion and say, "My Lord, be merciful to them as they have brought me up in my childhood." [Al-Quran, Surah Al-Isrā, 23-24]
As such, in these situations it is best to first make an attempt to reconcile and find a middle path that would please both parents instead of foregoing the request of one for the sake of the other. The Holy Prophet Muhammad (peace be upon him) greatly stressed the importance of giving respect to one's mother and maintaining good ties with her.
Consider the following hadīth (saying of Muhammad):
عَنْ أَبِي هُرَيْرَةَ رَضِيَ اللَّهُ عَنْهُ قَالَ: جَاءَ رَجُلٌ إِلَى رَسُولِ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ فَقَالَ: يَا رَسُولَ اللَّهِ، مَنْ أَحَقُّ النَّاسِ بِحُسْنِ صَحَابَتِي؟ قَالَ: «أُمُّكَ» قَالَ: ثُمَّ مَنْ؟ قَالَ: «ثُمَّ أُمُّكَ» قَالَ: ثُمَّ مَنْ؟ قَالَ: «ثُمَّ أُمُّكَ» قَالَ: ثُمَّ مَنْ؟ قَالَ: «ثُمَّ أَبُوكَ»
Abu Hurairah narrates: A man came to Allah's Messenger (ﷺ) and said, "O Allah's Messenger (ﷺ)! Who is more entitled to be treated with the best companionship by me?" The Prophet (ﷺ) said, "Your mother." The man said. "Who is next?" The Prophet said, "Your mother." The man further said, "Who is next?" The Prophet (ﷺ) said, "Your mother." The man asked for the fourth time, "Who is next?" The Prophet (ﷺ) said, "Your father."
If you are faced with a situation where your mother will be displeased by your decision to obey your father's request, then you must kindly engage her and never retort to any comments she might make in the state of displeasure. If she shows her disapproval by directly approaching you then you should observe patience and listen to her intently.
May Allah (God) give us all the correct understanding of religion and grant our parents success in this life and the hereafter. Aameen!