There's another car park on the coast road heading towards Eastbourne from Birling Gap. You'll see it on the right just pass Belle Tout lighthouse. Yet more car parking on the coast road just North of Beachy Head lighthouse and East towards Eastbourne. Personally I like the walk from Birling Gap.
Google is your friend! Car parking is shown on the satellite view in Google maps.
If you're thinking of walking along the beach, then Birling Gap is probably best, BUT it's essential you check tide tables and be aware that the chalk cliffs are dangerous, especially after rain and/or storms with a tendency to drop large bits on you.
Alternative possibility is a boat trip out from Eastbourne for a different perspective.