The Starry Rift, a collection by James Tiptree, Jr. (pseudonym of Alice B. Sheldon) comprising a framing short story, two novellas, and a novelette. This collection has its own Wikipedia page, and it was the (unaccepted) answer to the question "1980/90s scifi book about a librarian who recommends 3 short stories to two students at an alien university". The story you described in detail, about the girl who catches an alien parasite, is "The Only Neat Thing to Do". The descriptions below are quoted from an Amazon customer review by Rory Coker. The stories are:
"In the Great Central Library of Deneb University" (short story).
"The Only Neat Thing to Do" (novella, first published in The Magazine of fantasy and Science Fiction, October 1985, available at the Internet Archive):
In the first story, a teenage girl is given a "Space Coupe" as a birthday present, and shortly sets off on an unauthorized journey into unexplored space. She becomes infected with a mind parasite which is shortly going to eat her brain. The problem she must solve is getting the word back about the danger, without infecting the rest of the human race. Tiptree doesn't make me believe for one second that there could exist a planet where natural evolution could result in all mind and intelligence being carried by microscopic parasites, while all large animals are mindless, brainless hulks who function only when the parasites create brains for them. Or that the parasites must be carefully trained as children in "parasite day school" to avoid killing their hosts. That's not the point. Her aim is to take that bizarre situation as a given, and work out the logical consequences. In this, she is somewhat like one of her obvious influences, Murray Leinster.
"Good Night, Sweethearts" (novelette, first published in The Magazine of Fantasy and Science Fiction, March 1986, available at the Internet Archive):
The second tale is more conventional space adventure. A solitary man who makes his living salvaging space junk finds himself in a situation where he must rescue the most beautiful woman in human space (and her clone!) from insanely vicious space pirates. Tiptree's delight in describing the mechanics and details of her imaginary interstellar space ships, space communications, and resulting space puzzles to be solved by the characters, is obvious throughout.
"Collision" (novella, first published in Isaac Asimov's Science Fiction Magazine, May 1986, available at the Internet Archive):
In the final tale, explorers encounter a hitherto-unknown, technologically advanced galactic empire of three-sexed kangaroo-like aliens. The aliens are religious fanatics who practice sacrifice of their own kind, and who have good reason to destroy any humans they encounter instantly and with no questions. [It's those pesky space pirates, see story 2, again.] The explorers face the problem of avoiding an interstellar war, by somehow making the aliens understand that the human race doesn't consist entirely of pirates. This story is the only one of the three that contains fantastic/supernatural elements, and they don't help either the reader or the plot in any way. Again, Tiptree can't make the three-sex system plausible, and that's not her goal.