Using cover designs as a reference, I rarely found the Dragonriders of Pern to show males and fire lizards (tiny shoulder sitting dragons) on the same cover.
But if we're talking about an old series which showed a male, in a science fiction setting, with a dragon-like pet, Flinx and Pip have more covers depicting the requested elements.
Pip and Flinx have been around since the early 1970s.
scifi.stackexchange.com/users/2765/Thaddeus Howze
It's a pretty common idea.....
If it were Fantasy, I'd guess the Earthsea novels; they are called harekki , and an image that sticks with many people is of one of the side character's little sister with one that curls around her wrist.
And, of course, recently, people have become familiar with Daenerys Targaryen's little ones.
Since you say science fiction, tho, could it be from one of the Pern books?
The well known dragons started as little Fire-lizards and were genetically engineered for greater size, eventually becoming the dragons everyone is familiar with.
Take a look (at your own risk; it's TvTropes, and you can lose hours of otherwise productive time there) at the TvTropes page on Shoulder Sized Dragons .
One of them might be what you are looking for.
scifi.stackexchange.com/users/2292/K-H-W
Dragonrider's of Pern Series by Anne McCaffrey.
The small ones that sit on the shoulder are called Fire Lizards
scifi.stackexchange.com/users/26195/Dijkgraaf
Pip & Flinx by Alan Dean Foster?
scifi.stackexchange.com/users/31066/Patrick Wynne
Dragon Drums has some versions of the cover where a young man has a dragon (fire lizard) on his shoulder.
If you don't remember space ships I'd go with this one.
If there are ships it may be the Pip and Flinx series by Alan Dean Foster.
It has a young man with a flying snake on most of the covers that can be miss remembered as a small dragon.
scifi.stackexchange.com/users/47078/Amy
Most of Andre Norton's protagonists have animal sidekicks.
Forerunner, one of her books I read recently-enough to remember a bit, has ones more bat-like than dragon-like.
She wrote more than sixty books, so I think there's a good chance of a shoulder-dragon in there somewhere! ...
But, OTOH, a quick look at all the covers of her books that I could find on Bing, showed no such shoulder-dragons.
scifi.stackexchange.com/users/47116/Donald Heskett
An old series that had a young man with a small dragon that sat on his shoulder. Not a magic type book, more science fiction. I can't believe I can't remember the author.  Thanks.
Science fiction book where the protagonist has a tiny shoulder-sitting dragon
scifi.stackexchange.com
<story-identification><dragons>
scifi.stackexchange.com/questions/92887
scifi.stackexchange.com/users/47077/Jack Stratton
Suggestions include Pip & Flinx, the Earthsea novels, one of the Pern books (possibly Dragonriders), or Dragon Drums. The Pip and Flint series by Alan Dean Foster or the Pern series by Anne McCaffrey are the most likely choices in the genre of science fiction. However, in the genre of fantasy, the Earthsea novels are a frontrunner. Finally, a young man with a dragon — or a fire lizard — on his shoulder is depicted on some of the front covers of Dragon Drums.
It is thought that the book may be from the Pip and Flint series by Alan Dean Foster or the Pern series by Anne McCaffrey. The genre of science fiction encompasses both the Pip and Flint series and the Pern series. In the genre of fantasy, one possibility is the Earthsea novels. Some versions of Dragon Drums depicts a young man with a dragon — or a fire lizard — on his shoulder.  