These days, you often have separate companies producing the films and the big studios distributing them. But back in 1982, that just wasn't the case, and for a franchise at the tipping point like Star Trek was, Paramount wasn't about to let it out of their sight.
The studio itself ranks as one of the earliest ever founded in Hollywood, starting out in 1912 as Famous Players Film Corporation and eventually becoming Paramount formally in 1927.
Historically, they avoided becoming affiliated with one single genre, like other studios tended to do. (Warner Bros, for example, became known for its gangster pictures, while MGM had musicals, Universal had monster movies and Disney had…well, we're pretty sure you can guess what Disney had.)
Instead, Paramount banked on stars, pushing early screen icons like Mary Pickford, Rudolph Valentino, and Douglas Fairbanks as the draw for their movies. The formula worked very well for a time, though the company ran into trouble in the 1950s and 60s when their business model began to fade.
They were saved by a couple of things. First there was the arrival of actor-turned-producer Robert Evans, who restored their fortunes in the 1970s with a string of hits like Rosemary's Baby, Chinatown, Love Story and The Godfather. The second was their investment in television. They were early adherents of developing programming for TV and in 1967 acquired the rights to Desilu studio (Lucille Ball's company). That, in turn, gave them the rights to Star Trek, which is how Captain Kirk and the gang ended up in their wheelhouse.
For Star Trek II, the studio wasn't interested in messing around. The first Star Trek movie had gone way over budget, and while it made money ($82 million), that high production cost ($35 million) cut into the profits. (Source)
They booted series creator Gene Roddenberry, who they blamed for the bloated production, and brought in TV producer Harve Bennett to help this one.
Bennett came up with the original story along with screenwriter Jack B. Sowards. At the suggestion of fellow Paramount exec Karen Moore, he brought Nicholas Meyer in to direct. Meyer hadn't seen much Star Trek, but his Sherlock-Holmes-meets-Sigmund-Freud novel The Seven Percent Solution had the same funky reboot vibes that Paramount was hoping to find.
On a budget of $11 million, Bennett and Meyer delivered a winner to the studio. The film made about the same as the first one, but with a much smaller budget, the profits were much more profit-y, and Paramount was willing to go all in on more Trek movies.
The rest, as they say, is history. Paramount produced a whole slew of new films, as well new TV shows and attendant piles of ancillary products that brought the money rolling in. Trek remains one of the tentpoles of the studio, thanks largely to the success of this film, and while Paramount has a bevvy of great pictures to claim, this one holds a significant spot among them. (Source)