Not only bridges of that age - this has been done to some of the overbridges on the M4 (built c1969-71), although I think the habit of some who parked on the "pavements" on the bridges to sit and watch the traffic go by did not help.
Its a long time since I was involved in bridge design (about 1980) but
IIRC▸ the carriageway is designed to a full traffic load including an abnormal vehicle, but the worst loading case for a footway was a single accidental HGV wheel load of 113 kN (about 11 tons).
Unless the bridge deck had badly deteriorated I am not sure that a parked car would have much effect, but a parked multi axle HGV might as it could have more than one wheel load acting at once. I suggest that the barrier is most likely to be to keep HGV's off the footway therefore.