What does the phrase "a vector parallel to a plane" means

Is this the same thing as saying a vector that is on the plane?

Yes.

As long as you don't consider vectors to have defined positioning of their origins, those are exactly the same terms.

The only reason they use the word parallel, is to remind you that the vector isn't definitely part of that particular plane. It could be offset in its origin.

For instance, as I walk around the floor of my house, you can identify my displacement vector as being parallel to the plane of the ceiling. However, it is kind of confusing if you identify my displacement vector as being on the plane of the ceiling, because I never go on the ceiling. I remain on the floor.