I think there is no park here that allows you to BBQ. . . not sure though but i havent seen any :(

Corniche have these mini playground for kids.
Katara is nice for sightseeing.
You can BBQ at the beach but its a bit far from AlSadd though.
Aspire park have a running path/track where you can jog.

Now, to answer your question, i havent seen any park here where you can do all that, especially the BBQ thing :(