Well the station is very close to the centre of Highbridge (tiny place) so you could get a bus to BoS then bus to Brean, could be bit iffy with luggage and a bit of a phaff though.
Years ago I worked in BoS when I didn't have a car and tried going by train and walking. The walk was about half an hour. Then Brean is prob another 4 miles away. Easy by bus again.
You may be better getting off the train at weston and getting the bus to Brean. Our buses here are First Choice so you could look up some timetables?? Or look up local taxi firms and try to book a 7 seater?? Or even a taxi from weston?? Taxi rank outside or opposite is huge Tesco and also couple of good local firms (Apple, Arc, Central) you could prob pre-book?
Using the journey planner on Pontins website we would need to go by train to WSM. Then catch a bus which take 32 mins. Then a 3 min walk to the park. Hoseasons may be ok. There is one in Weymouth.Just reading up on it. Price is a bit more though.