Yes, you can paint over varnish. Benjamine Moore Paints makes a product called Fresh Start. You can put oil or latex paint over it as your finish paint. ANd, if I'm not mistaken, this product doesn't require any sanding. It's like $20+ a gallon.
Be SURE to clean the surface first. Talk to BM about what to use to clean it first. My best to ya and hope this helps.
Jay J -Moderator
PS: The same product can be used if the existing finish is Polyurethane ...