My kids got in to the nail polish and decided to paint the table. Go figure.
Is my only resort to sand the table?
Acetone, i.e., nail polish remover. It may not remove it fully if the table didn't have a sealer coat on it and the polish went down in the pores of the wood.
Prepare to at least touch up the finish on the table.
Acetone (or any of the other chemicals used to remove nail polish) are likely to strip the existing finish or at least damage it pretty badly.
As W.C. Fields said: "I like children, if they're properly cooked."
Best bet may be to turn the piece over.
Teak responds well to a light sanding with the grain using fine sandpaper, that should take off the nail polish. You'd need to do the whole tabletop so that the finish is uniform. Put the kids to work helping, it'd be a good lesson.
There are enough little spots on the table (which are inevitable) such that it wouldn't be to soon for a light sanding or cleaning anyway.
Paint the table blue, They stayed between the lines fairly well at least. Sorry couldn't resist. I like snidleys fix.
I don't know about making them help as a lesson, if they like sanding they might start sanding everything you own!
The worst kid damage I ever saw was when a friends kids played school and used his recently imported from Japan 70" TV screen as a black board with magic marker, no fix for that.