Is there a word for somebody who cares for his/her child too much?

Solution 1:

An entertaining neologism for this is helicopter mom or helicopter dad. (The implication is that the parent is "hovering" over their child at all times.)

Solution 2:

You could call such a parent over-protective. Sometimes such parents are called intrusive or smothering (in the sense that their habit of getting between a child and the real-life ups and downs of experience may stifle a child's emotional growth).

Solution 3:

Overbearing (definition from Merriam-Webster)
: often trying to control the behavior of other people in an annoying or unwanted way

Solution 4:

One classic phrase is a 'doting parent'.