When we visit my mum occasionally we pop down to this park. Its a huge green area of land with woodland on both side, leading down to the pebbled beach. Once you get to the beach and if at the right time you can walk up to Barry Island or down towards Fontygary beach. Either way, you can just walk and walk
The parking is excellent and they have toilet facilities and a couple of doggy poop bins too.
Its funny too cos whilst you meandering down the green, you can watch the planes fly into Cardiff Airport above your head, as the park is on a lower leval to the runway

Don't worry though, its not that noisy
MapThis site explains it better than I could
Porthkery Park