Yes and no. Dark colors have a lot of tint and some of this tint can migrate into the newly applied layers of paint, changing the final color a little. A good primer has extra binders, these binders will seal up the underlying color so its tint won't migrate. The only problem is the primer doesn't have a great deal of tint so it won't cover as well as paint.
You will be applying 2 or more coats of semi-gloss paint to cover up the red, could be 3-4. The choices you have are; apply 1 good coat of primer and possibly 2 coats of paint or keep applying the paint until you get good coverage. If you were to use a flat or eggshell paint I would suggest not using a primer but semi-gloss has low coverage so a primer will help.