Pulling wire using previous wire?
Your odds of success are about 1% unless the wire in in conduit, and it almost certainly isn't.

Generally, wires in walls are stapled to the framing. Perhaps your thermostat wire isn't, and then you might have a chance, if it's a straight shot. If there are staples or the wire goes around corners, you probably won't be able to pull the old wire out, and if you do manage that, the point where the new wire is connected to the pulling wire will not be happy about going through staples or around corners.