Are “The Dirty Pair” actually a part of the “Crusher Joe” universe? Or are they just a film parody in that universe? The worlds of The Dirty Pair and Crusher Joe seem to overlap. But it's never been clear to me if the The Dirty Pair (aka: "Lovely Angels") were actually a part of the Crusher Joe universe or were they just a parody entertainment film in the Crusher Joe universe?

For example, in this clip in the film Crusher Joe (1983), the team is clearly watching a film of the The Dirty Pair. Is this all just a layer of meta-reference?
Yes, both Dirty Pair and Crusher Joe are in the same universe/continuity.
Aside from the fact that both were written by Haruka Takachiho, Dirty Pair took place around 20 years before Crusher Joe.
From Japanese Wikipedia on Dirty Pair,

This work which was also written by the same author who had written Crusher Joe shared the same universe and linked to Crusher Joe 20 years later.

Dirty Pair OVA series review by Otaku USA Magazine also states

The series was created in 1979 as a set of novels by Haruka Takachiko, starting with the publishing of Great Adventures of the Dirty Pair, released in the U.S. by Dark Horse comics. The novels were done in connection with another of his works, Crusher Joe, [...]. Crusher Joe is similar, though a bit more serious, fare about a group of galactic mercenaries who have their adventures a couple decades after the Dirty Pair.

Lastly, from TV Tropes on Dirty Pair (Warning, TV Tropes link),

Novelist Haruka Takachiho first created the Pair in 1979, in a series of stories published as Light Novels set in the same continuity as his other series Crusher Joe (which debuted 2 years earlier).

Crusher Joe article on Wikia  described the connection in more detail (in accordance with the Japanese Wikipedia counterpart):

It takes place in the same general universe as the Dirty Pair light novels with cameos between the two franchises.



 The two series officially crossover in the light novel Dirty Pair's Rough and Tumble where Kei and Yuri meet Crusher Dan, the father of the series protagonist Joe.



In the Crusher Joe: The Movie, A movie of the Dirty Pair is shown at a scene taking place at a drive-in movie theater.



 In Dirty Pair: Project Eden, the four main characters are shown briefly on the monitor in the Lovely Angel while Mughi is controlling it.


Apart from that, the drive-in Dirty Pair movie on Crusher Joe was adapted from The Great Adventure of the Dirty Pair, the first series of the light novel.
From Japanese Wikipedia on Dirty Pair,

The first appearance was in Crusher Joe movie's scene on The Great Adventure of the Dirty Pair movie.

From Japanese Wikipedia on Crusher Joe

In the middle of the movie, there's a scene where Joe watched Dirty Pair movie. It's the movie about The Great Adventure of the Dirty Pair[10].
[10] From The Great Research of the Crusher Joe, page 118 about &quot;Drive-in theater&quot;.