For some, it may not just be about reframing birth control (while nice for some of the period type symptoms/issues) into a different thought process.
I had always had severe reproductive issues (endometriosis, pcos) where the standard treatment is birth control. I wasn't out, nor even fully aware at that time that trans was a thing so I went on it as a teen quite willingly as I was miserable physically. I never did find a formulation that didn't make my brain feel like it was on fire. Not literally, of course, but I couldn't stay on them long as the psychological symptoms were off the chart for me, even in low dose, even on multiple formulations. I never tried depo but that was about the only one I didn't try before I started T and it no longer mattered.
I know the OP has just started them and it may be possible that reframing the situation (OP - being on birth control does NOT make you less of a man!) will help. But it is also possible that his body/brain just rejects those types of medications. Obviously, it was his first dose, it is too soon to tell.
I think it is equally that he is given no control over the situation and no one (teen specifically) should be forced to take medication unless it is medically necessary. I didn't see it in the OPs post, but if his parents are forcing this on him as a means of control, not because his physical symptoms require it, that is wrong.