Question Summary:
I deeply regretted this sin and ended up admitting to my father what I had done and paid him back the money

Question Detail:

When I was 13 I was given some money from my father to pay off my Islamic school fees. Unfortunately I forgot to give the money and decided to keep it for myself. I deeply regretted this sin and ended up admitting to my father what I had done and paid him back the money. However I did not give any money to my Islamic school nor did they ask for it. I felt that I only owed my father the money. But I'm wondering, do I owe them money as well? 

Answer :

In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
Imaan is the strongest support for a believer. It is the barakah of your Imaan that you resent the wrongs you have committed and wish to reform. May Allah Taala grant you steadfastness in Deen. Ameen.
In principle, the school fees should be paid to the school. Discuss this issue with your father, and express your concern to him that the school fees ought to be paid to the school. If your father is unable to return the fees to the school, you may make an equivalent amount as donation to the school. 
And Allah Ta’āla Knows Best
Huzaifah Deedat
Student Darul Iftaa
Lusaka, Zambia
Checked and Approved by,
Mufti Ebrahim Desai.
____________________
مجلة الأحكام العدلية (ص: 105)
(الْمَادَّةُ 562) تَجُوزُ إجَارَةُ الْآدَمِيِّ لِلْخِدْمَةِ أَوْ لِإِجْرَاءِ صَنْعَةٍ بِبَيَانِ مُدَّةٍ أَوْ بِتَعْيِينِ الْعَمَلِ
مجلة الأحكام العدلية (ص: 105)
(الْمَادَّةُ 568) لَوْ اُسْتُؤْجِرَ أُسْتَاذٌ لِتَعْلِيمِ عِلْمٍ أَوْ صَنْعَةٍ فَإِنْ ذُكِرَتْ مُدَّةٌ انْعَقَدَتْ الْإِجَارَةُ عَلَى الْمُدَّةِ حَتَّى أَنَّ الْأُسْتَاذَ يَسْتَحِقُّ الْأُجْرَةَ لِكَوْنِهِ حَاضِرًا وَمُهَيَّأً لِلتَّعْلِيمِ قَرَأَ التِّلْمِيذُ , أَوْ لَمْ يَقْرَأْ.