Is the repentance of a parent from major shirk accepted, even if he claims he committed it out of ignorance and not intentionally? Is it to be understood from this that he does not regret it? And what is the correct meaning of regret?

Supplicating to other than Allah the Almighty is major shirk (polytheism), and one who is ignorant of its ruling is excused and not judged to be a disbeliever. If the father repents, acknowledges his error, and refrains from supplicating to other than Allah, then his repentance is accepted, Allah willing. His declaration of ignorance does not diminish his remorse; rather, it indicates the sincerity of his repentance. Remorse is a great pillar of sincere repentance.