June told Beaver that in Season 4 episode 9 Beaver's IQ. That would have had to have been June's father as Ward's father was a dairy farmer.

I thought June was raised by Aunt Martha. Were June's parents ever mentioned outright?

Someone might have a better memory than I, but I don't recall any reference to June's parents.

I think in the Visiting Aunts Ward explains to the boys that Aunt Martha raised June. What we don't know is why, and what happened to her parents. It makes sense that we don't have great details, since LITB is centered on the boys' lives.

June’s mother wanted June to be an actress and got her acting lessons. Ward was asked to come to dinner often by June’s mother. Of course June’s mother made June get up in front of the whole school to say she lied that her mother was a movie star named Laverne Laverne.

Now that you mention it I do remember the last reference.

I guess she never mentioned her father, and how and when she ended up in Aunt Martha's care.