No one does al haesh in the park in English. The answer shouldn't even be allowed. It's also still not accepting "have a barbecue".
"Every Independence Day they have a BBQ in the park." They are doing is less grammatical. If its a habit once a year, the simple present forms, "they do/they have/they make/they host" are appropriate.
Also, ‘they have’, not ‘they do’.
Why not ‘they make a barbecue’?
Not in English. That would mean something different like constructing a barbecue rather than using a barbecue to cook food which is "have a barbecue".
What about a "cook out"?
Not necessarily. Like making lunch
We often make a barbecue, no construction involved.
Yes. In America we often say "making a BBQ."
I would omit the ה in יום העצמאות. We're not discussing a specific day, just a date
Also why isn't Al Ha esh translated as barbecue? It is definitely not a word for which there is no English equivalent. And Independence Day barbecues were not invented by Israelis!!!