You might check with an attorney in your area, but generally in the US, unless those covenants were filed/recorded with the property records clerk or a court, they are not legally binding. They are just an agreement between neighbors, and you as the new property owner did not agree to it. For example, I bought a house and was told by the previous owner that my neighbor will not mind if I cut through their property to get to the lower part of my property (an easement basically, there's a road behind their property that connects our properties, old farm track). That's an agreement the neighbors had between them, but it's not legally binding and I would technically be trespassing but nobody enforces it.
HOA covenants are usually recorded with the county and would have been presented to you at closing along with the title search. You would have had to sign them.
In the short term, I would recommend, fence in your poultry or build a large enclosed run to protect them. The easiest thing would be a moveable electrified poultry fence. Your birds will be safer from ground predators, including neighbors. It'll cost you around $300-$500 to set up a mobile poultry fence from Premier1.
Check your property records as to the zoning of your property (look for a a GIS website) and then call the county land use office as to what is or isn't allowed in that zone. They will tell you straight up what the ordinances are as to chickens and roosters and other livestock, i.e., what you can and can't have.
If, after arming yourself with the above information, the neighbors persist, I might consult with an attorney to possibly draft a cease and desist letter.
