In a word, yes. I sometimes use varnish as a primer under paint if I haven't any primer about.
Check that there are no whiteish patches in the varnish - they indicate that the varnish has parted company with the substrate, and need to be cut out and patched first. Wash the varnish down with detergent to clean & degrease it, then rub down well with a reasonably coarse paper to give the paint a good key.
Commonly done by wooden boat owners taking their boats into tropical waters for a short period; no problems if the varnish is in good nick. In fact a good idea, because the process is reversible, whereas once you paint the bare wood, that's it for ever.
If you want the paint to last for atleast one season, do not paint over varnish. Paint sits on top and will not bind into varnish. It will scratch and flake quickly!
it will work under some circumstances, provided youy prepare properly - sand and clean. BUT - if you apply polyurethane over conventional or vice versa the bond will be poor.