I'd imagine that growth spurts of the horn are painful for a young unicorn. Maybe it's akin to a toddler getting their first teeth. Poor Twi's widdle head hurts so much. Good thing the princess is there to offer comfort.
We saw Twilight's parents in Cutie Mark Chronicles. They clearly support(ed) her academic pursuits and seemed to be happy with her. Beyond that, there isn't really any evidence suggesting the degrees to which they and Celestia raised Twilight. I imagine it would have been a mix of both, perhaps leaning more towards Celestia.
I meant from that point onward, them leaving her in the care of Celestia and her never ever mentioning them other then to the CMC's (and that was like "oh i lived with my parents until i became Celestia's student, but you dont want to hear about them")
It is curious, isn't it? Maybe her relationship with them isn't sunshine and rainbows. It's entirely feasible, given what we saw of typical Canterlot citizens in Sweet and Elite, that they were all too willing to give her to Celestia and bask in their own pride.
Twilight isn't your typical Canterlot noble snob either, maybe her parents weren't that wealthy, more like poor, so being semi adopted by the ruler of all ponies is a jackpot and they let her have her perfect life with the princess.
Her fear of disappointing Celestia is like that of a child towards its parents.
All this will be shattered with the season finale episode of course, Twilights parents will surely be there.