Going out on a limb here, I would say that open-fires, and therefore barbecues, are not allowed in public parks in Athens, or anywhere else in Greece for that matter. This statement is based on the fact that Greece is a country at a constant risk for wildfires during the hot and dry season, which is the peak time for barbecues. I could not find any information (in English) specific to public barbecues. I did however find some statistics behind the wildfire claim. Taking the most recent, most devastating event, in 2007, wildfires consumed circa 670,000 acres of land.
It is also worthy to note that wildfires don't happen only in national parks. In 2007, several suburbs around Athens were involved in the catastrophic summer wildfires.
This is not to say that the Greeks are not barbecue fanatics. Indeed, most houses with a garden will have either a brick barbecue with an automatic spit-turning system to make Kokoretsi, or a dedicated area for a mobile barbecue.