If our parents wake up a little late for Fajr but I am determined to pray on time, would it be permissible to pray at a different time than my family, or is it not recommended?
as salam alaikum
if by "little late" is meant that your parents miss the best time for prayer (waqt al-fadilah), meaning just as enters the adhan time, there is no problem in praying at different times as long as everybody prays before sunset. However it would be appropriate to pray Jama'ah and even better to pray Jama'ah at the mosque if there is the possibility.
With prayers of your success.