The Universe is everything that exists, all of time and space across all dimensions, and all its contents.
Of course parallel universes where Rose has a dad or Spock has a beard aren't in the universe. All of reality, which is what Davros' Reality Bomb was going to destroy, is all the parallel universes, which is called the multiverse. It's like sheets of paper in a stack, with slightly different things printed on each sheet, but the stack basically it's all one big thing. And that one big thing, the multiverse, is everything. Except in the comics, where there's a stack of multiverses called the omniverse. But that really is everything.
Well, except for E-Space. That's not part of the universe, and it's not parallel either, it's green. Each parallel universe is called N-Space and has its own E-Space. Except when N-Space is something completely different, which is also not part of the universe.
But anyway, there are parallel universes and E-space negative universes, but there's nothing actually outside them. Well, except House's house, which is a Hotpocket Universe outside the universe, like the place where the Time Lords hid after the Doctor pretended to destroy their planet (as opposed to the time he actually did destroy it). In fact, a Zero Room is a hotpocket universe too. Oh, and the Divergent Universe is not parallel, it's… divergent.
Also, everything that happened before one of the Time Wars was actually in a universe inside a bottle inside the universe where all the later stuff happened. Not that it matters, because the entire universe, all of time and space, later ended and got rebooted out of Amy's memories. But forget that bit.
So, there are different universes in weird extradimensional relations to each other, but there aren't multiple universes in space, because the universe is all of space. Except there are Krakens and other things that swim between the universes. And there's also the Void between universes, which is different from where the Krakens live, because nothing can live there. Adam Smith comes from there.
The universe started with the Big Bang, aka Event One, the time at which time itself started, so nothing could possibly be before that. Except for The Beast. And the White and Black Guardians, who were the last surviving Time Lords in a universe before this one. And… actually, lots of other things.
And then there's Mictlan, which doesn't even exist except as a concept, so it's not part of the universe either, even though it's a place.
So the universe isn't actually anywhere near being everything that exists, despite that being the literal definition of the word. But still, it's pretty big. You could easily lose your car there.