Is it permissible to fast as repentance for unknowingly cursing one's parents?

What you committed is a great sin, but repentance expiates what came before it. You must feel remorse and resolve not to return to it, and increase good deeds, draw closer to Allah through supererogatory acts of worship, and show kindness to your parents.

You are not obliged to fast what you imposed upon yourself unless you uttered a formula indicating a vow (nadhr), in which case you are bound by what you committed yourself to.

The jurists have stated that whoever vows to fast for an entire lifetime (dawm ad-dahr) is obliged to fast it, with the exception of Ramadan, the two Eid days, and the Days of Tashreeq; these days are not to be made up. Making up missed Ramadan fasts and expiations for Dhihar, killing, intercourse during Ramadan daytime, and oaths takes precedence over the vow. If one breaks the vowed fast with or without an excuse, they do not have to make up what they missed, but an expiatory payment (fidyah) is obligatory if they break it without an excuse.