This story is probably not true. It sets Parker up as her own straight person, and that was not her style.
When looking for a room, she did state “Nothing fancy. I just need a place where I can lay my hat and a few friends.”
When she registered as a single woman in a hotel and then brought a man up to her room (horrors!), the fron desk called her and asked “Do you have a gentlemen in yur room?” Supposedly she replies “I don’t know. I’ll ask him.”
At a party, she met an American actor who had just finished a season in London and more than once referred to his “shedule”. She said, “If you don’t mind my saying so, I think you’re full of skit.”
Dorothy Parker and Alan Campbell were wed in 1935, divorced in 1947, and re-web in 1950. They invited the same people to the second wedding as the first.
At the second reception, a guest remarked to her that “Most of the people in this room haven’t spoken to each other in years.”
Dorothy responded “Including the bride and groom.”
By the time you swear you’re his,
Shivering and sighing,
And he vows his passion is
Infinite, undying -
Lady, make a note of this:
One of you is lying.
The one I like that’s been attributed to her is that she said, supposedly on her honeymoon, in response to an editor’s query about her work that was due:
“Tell him I’ve been too fucking busy. And Vice Versa”
I’ve never seen this one attributed, and it has an air of Too Good To Be True about it. Wikiquotes agrees about the lack of provenance:
Incidentally, as I pointed out recently, while it’s true that Parker, in reviewing A.A. Milne’s The House on Pooh Corner wrote “…Tonstant Weader Fwowed Up,” that’s not the entire review – it’s just the last four words of a much longer piece (but they’re the ones that everyone remembers, or quotes because it’s all they’ve heard.) This one, however, I know she certainly wrote – the entire review is in the collection The Portable Dorothy Parker.
You Might As Well Live, a good bio about Pawrker, agrees that the “fucking busy” quote is probably not true. It more likely would have gone like this:
Boss: Why haven’t you finished that article?
Newsman: I’ve been too fucking busy.
Parker, sitting nearby, in a tiny voice “or vice versa.”
Trivia: Parker and second/third husband Alan Campbell had a very Norma Desmond/Joe Gillis type of relationship. The actor who played Joe Gillis during Sunset’s Broadway run was named Alan Campbell.
Just so everybody’s on the same wavelength, the “this” being referred to is the quote in the OP: “Heterosexuality is not normal, it’s just common.”
If Parker said it, she undoubtedly said it long before Derek Jarman, just from their birth and death dates. But I can’t find any evidence of anybody at all saying it before Jarman’s At Your Own Risk: a Saint’s Testament, published posthumously in 1992. (He died in 1986.) So I agree he should get the credit and indiscriminate internet attributors the blame. Good catch.