The parents share the responsibility?
In today's world, yes. In the 50s and 60s - the parents had no reason to suspect. It may seem naive but the children who were chosen to help out in the rectory or on Saturday in the church were the "good" little children. Their parents were thrilled that they were chosen.
I often helped out in the rectory when the priests needed mailings done or gifts wrapped. I always helped clean the church and get it ready for Sunday mass.
Luckily, the one priest in our parish who was a paedophile preferred little boys and I only found out about that years later.
My brother knew and always refused to have anything to do with that priest. My brother was considered a problem child. He never told my parents about the priest's actitivites because he didn't know how and they would not have believed him.
The priests of my childhood were almost all veterans of WWII. They served on the front lines as chaplains. They were good decent men. One of them often came to visit us for a meal or just conversation. We trusted them.