For higher magnification target scopes (say > 10X) it is highly desirable, for rimfire and centrefire alike, to have an adjustable objective, so that parallax can be dialled out.
Oftentimes scopes with adjustable objectives that are intended for centrefire rifles, will not allow adjustment of parallax closer than 50-ish yards. You want a scope that will adjust down to 20 yards, and possibly even a bit closer. Presumably scopes labelled "Rimfire" would include this capability.
For scopes without parallax adjustment, they are set at the factory to be parallax-free at one distance. "Centrefire" scopes are usually set up to be parallax-free at 200 yards, "shotgun" scopes to be parallax-free at 100 yards, and "rimfire" scopes at a closer distance (not sure if 20 or 50 yards is typical).
You certainly can use a "centrefire" scope on a rimfire rifle and vice-versa. These labels are usually used to denote that the particular features of a scope are suitable/applicable/desirable for XX-style of shooting.