It's a question of subordination or parallelism between clauses. You can't make the act of asking Dad parallel to the obligation to stay home, it would imply the obligation was realized at the same time you asked, not after he answered.
Same thing with "I had to stay home and study." Here, to stay home and to study are parallel and contemporaenous. "I had to" can't be matched in tense with "study" because this breaks the rule that only the first verb in a compound verb is tense-conjugated: I had to study, not I had to studied.
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