I think you could get across the park to Barry from looking at the maps. Below some comments from sites which draw attention to some difficult parts which you may have problems with. You may have to get your bike over a stile in a fence - this should be no problem if you can get your panniers off easily.
"Apart from the one access road, which is heavily 'speed bumped', traffic is not normally a problem, although obviously care needs to be taken with small children and pets. The road continues beyond the car park, and under the viaduct, but is closed off by a barrier. This is probably a good thing, as it is not a good road! The track continues north to the Egerton Grey hotel, and then joins the airport road south of the main terminal".
barrywales.co.ukporthkerry.htmOpen ↗
Aeriel photo of area
multimap.comphoto.cgiOpen ↗
Comment from website
"Now let’s move on. Head east towards Porthkerry Park, a track in the woods, close to the cliff’s edge. Beware, take light steps or you might slip on the dark mud. Enter the park of Porthkerry, once owned by Romilly Estates. This large park becomes water-logged in winter, a sign that once this large sheltered valley was an inlet to the sea. You will now be aware of a large viaduct, built at the end of the last century. The park must be negotiated with care as it is easy to become lost. Head east along the open grassland, with great woodland on either side. Maybe you might see the ghostly figure of Annie Jenkins, the 18th century witch of Cliffwood Cottage. Or even still the ghosts of smugglers may leap out at you and question your presence in their domain. You will enter the second part of the park under the bridge. Continue for 50 yards until the roadway forks in two. Head north along the roadway that soon becomes unmetalled. Go past a house on your left and over a fence. You are still on the footpath. Keep your eye on the ground, wary of outlines of buildings which once belonged to a medieval village of Cwm Ciddy. As you head north several gates block your way: negotiate these. You will see a large complex of newly refurbished building on your left, a model farm maybe?
Continue north along a roadway for ½ mile until you reach the A4226. This road is the main access route for the airport - due to change. Please note that the new Airport link road will be developed over the next few years, this description does not take that into account. Follow the road east for just over ¼ mile until you reach a traffic island. Continue on over this into the northern part of Barry. This may be a good point to break your journey for the day. It is possible to walk, get taxi or public transport into the centre of Barry"
celticway.orgvaleofglamorgan.htmOpen ↗
Cardiff Cycling Map
cyclingmapsonline.co.ukCyclingMap.aspOpen ↗