When I was raising my first child, I discovered a strange, apparently innate instinct which presumably is related to the mirror neurons:

At some point the child gets old enough that you start to feed them 'baby food' on a spoon. The child isn't used to eating off of a spoon so for many weeks the process is messy. The initial challenge is getting them to open their mouth wide enough for the spoon to enter. And telling them 'open your mouth' is not particularly useful since they don't understand English at that age.

Instead, the following instinct kicks in: as you approach their mouth with the spoon, your own mouth opens. They see your mouth open and then open theirs. The crazy part is that your own mouth opening happens involuntarily at the moment you want their mouth to open. It is physically difficult to suppress it, even if you try.

I've also noticed the reverse. When the child gets a little older, they at some point want you to open your mouth, because they want to feed you something or they are curious about the inside of your mouth (this is a phase they go through). They seem to also involuntarily open their mouth wide when they want you to open yours.