I spent 8 hours at Flamingo Land the other week before leaving reluctantly to catch the bus, but the park was due to close in ten minutes anyway! You do not need a full day to see everything, but I am a full-day zoo visitor whenever it is possible. I have not been to the YWP for two years, but it is easy to spend a full day there too.
From where are you planning to use public transport? Flamingo Land is served by regular (generally hourly) buses in both directions i.e. immediately from Pickering and Malton. The Pickering buses start in either Whitby or Thornton-le-Dale, whilst the Malton ones usually start in Leeds and travel via York but some start in Malton. The company is Yorkshire Coastliner. I am familiar with only the journey from Malton as I travel by train to there and then catch the bus to Flamingo Land (some go to the park entrance, but most only to the roundabout in Kirby Misperton from where it is a 5 minutes walk). The bus takes around 15 minutes from Malton.
I believe there are several buses to Branton from Doncaster, but on my three visits to the YWP I have travelled by train to Doncaster with my folding bicycle and cycled from there. I have noticed a bus stop near the lane leading to the pedestrian entrance.
If you are thinking of visiting both places on the same day and travelling between them by public transport I'd say it is not really feasible as the journey would take a considerable chunk out of the day. The quickest way would be to travel by train from Malton to Doncaster which would mean changing in York. If you were lucky with connections that should not take too long, although I am not sure just how long it takes from York to Doncaster - Malton to York is timetabled to be around 23 minutes. With the bus journeys at both ends I should not be surprised if it took up to three hours; it would definitely be more than two I think.
