Not so. West Yorkshire Road Car sold all its bus services off. Leeds and Bradford operations did go to Yorkshire Rider but those based on Keighhley, Harrogate and York/Malton went elsewhere. York was a few years later acquired by Yorkshire Rider.
No, you have it backwards!
West Yorkshire was privatised by sale to the AJS Group- essentially a management buyout, but with some complications which are not relevant here.
In December 1988 the Keighley, Harrogate and York/Malton operations were split from the main company (as Keighley and District, Harrogate and District and York City and District, respectively).
In July 1989 the "West Yorkshire" branded operations from Leeds, Bradford and Otley were sold to Yorkshire Rider along with the trading name. YR then had a number of buses in its livery with West Yorkshire fleet names.
AJS retained the rest at that point, all trading under their local names and with legal lettering to match.
At some point in 1989 AJS purchased York new-startup independent Target Travel.
In early -1990 Malton was split from York C&D to form Yorkshire Coastliner.
Then in July 1990 AJS sold York City and District and Target Travel to Yorkshire Rider, who also bought Reynard Buses at the same time.
So Yorkshire Rider ended up owning the West Yorkshire name, which presumably passed with it to First (eventually).
Similarly Yorkshire Terrier is mentioned up thread; that company was bought by Traction Group who passed to Stagecoach, so chances are the name is owned by Stagecoach!