Is there a chance my period will start while I’m still on the pink pills?

You don’t have real periods on the pill at all, ever, because the (combined) pill stops ovulation and thus you don’t have a natural cycle. Any bleeding you might get during your active pills is breakthrough bleeding. Any bleeding you might get during your break is just withdrawal bleeding which is a side effect of the drop in hormones. So yeah you may bleed whenever, but it’s not a period just breakthrough bleeding

You can bleed whenever