Parking information for the New Forest, Hampshire
There is a large free car park in Lyndhurst. However it apparently fills up fairly early in the day and was almost full when we arrived at about 11:00 am.
There are plenty of car parks dotted around the actual forest and you should have no trouble finding space. These are set back from the road and the entrances are usually marked with a small post as shown here. If you drive too quickly you may miss them as they are not conspicuous and rightly so.
The Ordnance Survey 'New Forest' map at a scale of 2.5 inches to the mile shows positions of car parks which is very useful.