Try removing the paint from a small section. Pick a spot that's inconspicuous (I can't spell) and see what you can do. Try to find out what the experts recommend for removing paint. However, they will need to know what the floor is finished with (poly or varnish). I bought a house 6 months ago with 1400 sqft of oak parquet floor. The owner sanded and touched up some spots and I cannot tell by looking. He pointed out where he did touch up work. He did not sand down to the wood because he never would have been able to match the stain. He screen sanded some bad spots and reapplied the same polyurthene (satin finish) that was used in the house originally.
I would start with hardwood floor cleaner to see if the paint can be removed without damaging the finish. And I would try to determine what type of finish you have.
Keep in mind that if a poly floor has wax on it (even overspray from endust) that another coat of poly won't stick. They say that those floors need sanded to wood and redone.
Even if you experiment on a section and cannot get the results you want. Then you can redo or have the floors redone. I would spend the $2,000 and have the floors redone professionally because I could afford to. Get it done while the house is empty. Most places charge $2.00 to $2.50 a sqft.
I would start by experimenting carefully to see if the paint can be removed without damaging the finish.