I think there's a good chance he's not lying about being embarrassed. He probably would be embarrassed because kids that age do get embarrassed about stuff like that.
You didn't ask this, but it's hard to answer without throwing it in. I think whether or not you should talk to the parents depends on the circumstances involving his "going to his friend's house". Does that mean if he's hanging out in the neighborhood and wants to sometimes go in the house to have a snack, or does it mean he wants to stay overnight at someone's house? Is it a new friend, or a friend he's had for quite awhile?
Either way, there's a good chance he's being honest about feeling embarrassed, but you could tell him something like, "You're just going to have to explain to your friend and his parents that I'm not your mother/father and feel a little more careful about taking care of someone else's child."
By the way, I have two sons and a daughter. Boys aren't really a whole lot different from girls when it comes to something like whether a thirteen-year-old would be embarrassed about some things.