Father Gonzaga, the parish priest, symbolizes the orthodoxy of the Catholic Church. When he discovers that the old man with enormous wings doesn't speak Latin—the language of the Church—or look pure and clean like an angel in a religious painting, the Father concludes he can't possibly be an angel.
Pelayo and Elisenda are symbols of the pragmatic, secular world which believes in superstitions, but is so rooted in the material they can only treat the miraculous as something to shove imperfectly into their physical realm. They treat the old man like an animal while exploiting him for profit as a freak show item. As he is flying away, Elisenda refers to him as an "annoyance."
Further, the iron bars that Pelayo and Elisenda put up on the windows of their big new house are expressly installed so "that angels wouldn't get in," a symbol of the couples' continuing resistance to the sacred. (Ironically, the old man does get in, symbolizing the impossibility of keeping out the sacred.)
The old, winged man is a Christ figure, as he is both human (bad smelling, bug infested, missing teeth, old) and divine (winged so that he can soar over the earth) at the same time.
The new feathers the old man grows when winter has ended are a symbol of rebirth. He uses them to soar into the sky (after a few struggles) and fly away.