* Mirrors don't actually flip images left to right. They reflect light, and that reflection creates a reversed image.
* The "flip" you see in a mirror is actually a front-to-back reversal. Your left side appears on the mirror's right, and vice versa. This doesn't affect how your hair is styled, as that's based on front-to-back orientation.
* Your hair parting looks the same in the mirror as it does in real life.
So, the answer is no, taking a picture in the mirror doesn't flip your hair the other way. It just shows you a reversed image of yourself, but your hair's style remains the same.