I remember watching this on VHS, it was an animated movie or something like that, with a girl elephant as the main character. All I remember is she was walking home and singing that "bringing home a baby bumblebee" song, and another part about singing about an old man's or goat's beard hair and things getting lost in it. :( It had to have been made in the 80's...

Could it be "Magic Adventures of Mumfie"?

Nope. :(

They sang that Father's Whiskers song... "They hide the dirt on daddy's shirt, they're always in the way."

There is a show that had that song but not in the intro it was "skinnamarink". Besides that I'm a blank sorry.

Skinnamarink t.v. it was an awesome show, the elephant was Ellie if I remember right, and there was a boy cat named copy cat.