A poor guess indeed! Francis Road in Hounslow West is the first parallel to Bath Rd, from where there are plenty of buses towards LHR (may need a change by the tunnel, but it will be very quick since there's a ridiculous abundance of buses as these routes are serving both the airport and other local traffic; Bath Rd is an A-road). H.W. tube station is about 7 minutes on foot and offers direct tube service to LHR.
Have managed to avoid LHR since 2006, but google maps are showing the street with plenty of cars parked and without any obvious markings showing (just don't block the access to anyone's driveway!), so I'm reasonably confident that free parking is still possible there.