I think Sakura's parents are a complete blank slate. I wouldn't make much of her complaining about how Naruto has it easy because he doesn't have parents because a lot of kids that age would be thinking more of the discipline that their parents would provide as a bad thing, not as a positive.
Inner Sakura could be a manifestation of a pair of domineering parents or it could be just a result of her being shy at school among classmates and have absolutely nothing to do with her parents. Since Inner Sakura seems to mostly come out around Sasuke (at least when not using fillers), I think it is mostly just because she's trying to be the perfect girl for Sasuke.
Actually, in the manga it came out when Naruto was around most the time, and sparked by his actions. Both the first and last times that happened in the manga are perfect illustrations of that.
As for her parents, I don't think Sakura was exaggerating- one only has to look at her personality at the time Ino befriended her. That's what kind of girl her parents had molded her into- weak, fearful, tearful, desperate for approval and to be liked. She's even more victimized than Hinata was. Hinata faced scorn from her father, but apparently had no difficulties in school, she played "invisable girl" and the other girls left her alone.
Sakura, on the other hand was tormented endlessly by her classmates. Her parents had to know about it, but did they teach her to fight back? Clearly not. It was Ino that taught her that. But somewhere, and we know it wasn't with Ino or at school, someone punished her for not being "the perfect girl". And from Sakura's own words we know it was her parents. Note, none of the other kunoichi have this issue, not even Hinata and considering how her father treated her, you'd think she would. Also note, Sakura's false properness is the most overt when an adult who she puts in a sort of parental role, to whom she thinks she must show she's a proper young woman. (So she's not judged as being the wrong sort of girl).
Dream you dream alone is only a dream, but dream we dream together is reality- Yoko Ono 1971
When you go to war, both sides lose totally- Yoko Ono
Remember, our hearts are one. Even when we are at war with each other, our hearts are always beating in unison- Yoko Ono 2009