We do bbq at the parks a lot in the cooler months. Most of the parks have special areas where you can keep your grill - Mamzar, Zabeel, Safa, Creek, Jumeirah... Just bring your food, chairs/table, carpet, bbq & bbq equipment :) Small tip though - try to find the spot a bit further from the bbq area unless you want to have a smell of bbq in your clothes, hair and nose :)
You don't even have to hire a chalet at Mamzar. When we went loads of families were just barbecuing all around the beach park (park/green area before you get to actual beach).
Ooooh it smells lovely in the late afternoon when everyone is cooking! YUM!