Parades are processions in the public streets, usually a public spectacle or as a form of entertainment. Macy's parade is internationally renowned because of a multitude of inflated helium balloon floats.
In the context of the song this means: "I ignore those wishing to befriend me [now that I'm famous]." Teste refers to testicle. To be left hanging is to desire contact but have it denied. Taking a left is avoidance.