So far as I know, unless your visitation states otherwise, you can certainlyhave your mother/father come pick up the child.
Unfortunately, it's likely to be an upset for the children. However, when your ex doesn't allow the visitation, you file contempt charges and take her to court.
Chances are good she won't release them that first time.
You really have a few choices here. You either pick them up late, quit your new job, don't seem them at all, or have someone pick them up for you.
The ex seems to be pushing you between a rock and a hard space. You visitation time, should be your visitation time. This is your child also. Is the ex also going to complain if your parents she the child at all, ever again?
Mine did. So it wouldn't surprise me.
After my divorce, my ex was refusing even phone calls from my parents to the children. She pretty much just wanted my entire side of the family to fall off the face of the earth. Oh, but there was me sending the money thing, for child support, etc that she wanted me to be around for.
Hopefully this will go smoothly. However, sometimes it takes a trip or 2 (or 3) to court in order to wake some people up.