Is the son obligated to lighten his acts of worship, such as night prayers (qiyam al-layl), at the father's request, knowing that the son sometimes prays outside the mosque?

If you have missed prayers after puberty, you must repent and make up all the missed prayers until you are reasonably certain that you have cleared your responsibility. You can refer to Fatwas no. 66593 and 46960 for clarification on how to make them up. If performing Qiyam (definition) al-Layl (night prayer) is difficult for you, or if your father's prohibition is out of compassion, then limit yourself to what you are able to do. However, if your father orders you to abandon the supererogatory Qiyam al-Layl, you are not obliged to obey him if his prohibition is merely due to foolishness, according to Fatwa (definition) no. 72252. Congregational prayer is obligatory for an able-bodied man, and some scholars hold that it is obligatory to perform it in the mosque for whoever hears the call (adhan (definition)). If you would miss the congregational prayer by not going to the mosque, then it is not permissible for you to stay behind in obedience to your father, according to Fatwa no. 31934. However, if you find a congregation elsewhere than the mosque, you may obey your father.