"El perro parece enfermo."
Me parece would be a more subjective thing. If you were discussing the dog and your friend would say, that the dog is just tired you could say: me parece enfermo, meaning in your opinion he is sick, but not to everyone. If you say, el perro parece enfermo everone agrees.
Parecerse (a) (as a reflexive verb) means that something is very similar to something else. Ellos se parecen (they are very similar), tu te pareces a tu hermano (you look/act/whatever like your brother)