Horses do play but it is much more of a rough housing situation. You can, however, do games with your horse. Parelli is all about turning work into play in terms of making the work you do into game form. By doing this you make the exercises more interesting. In terms of the head thing, that is your call. I allow my horse to rub his face (which is perpetually ITCHY) on my back, etc. BUT, he knows he is only allowed to do this with ME, he will not nudge random people or crowd the space of someone leading him in, etc. There are times when it is appropriate. For instance, (imagine if you will, a horse wiht the itchiest face in the history of horses, that is my horse!) After riding I loosing the girth all the way and then I go and unlatch the noseband and then he is allowed to scratch his face by rubbing on my back or by my scratching it for him. He would not just go up to someone and try this. I do not find it to be a problem because he does not shove me or anything like that. He was taught at a young age to be gentle and he is very gentle all of the time.
Some horses, however can play games in the arena with you. I have known some horses that like to play a soccer-type game with a big ball. They kick it and you kick it back to them and they get the idea. I have known other horses who play hide and seek where one stands in an area of the arena or behind a jump (obviously quite visible
and then the person "looks" for them and when they get close the horse tries to sneak away. It is sort of like tag but you only tag the horse...he doesn;'t tag you, obviously.
I would encourage you to not take your horses too seriously as some people obviously do. Horses are not all that serious, they are interesting and they are playful with one another, they are curious....it is good to think about them in more "fun" terms, it keeps you from making things boring for them.