No HOAs here on LI, but many local villages and towns, and most have ordinances that require boats, trailers, and RVs to be parked behind the front facia of the house, so that means side yard or back yard. Similar to ordinances that require you to mow your lawn, prohibit vehicles with no plates from being parked in the street, etc., the intent is to keep the neighborhoods looking nice while recognizing everybody has hobbies.
You really don't want this ordinance repealed because even though you will keep your nicely maintained trailer neatly parked in your driveway, folks will take advantage and you'll wind up with a neighbor that has a hot dog truck with no motor, no windows and up on blocks on his front lawn.
If you want to petition for an exception, you would likely be granted an extension on moving it, but not a permanent exemption. It would be reasonable to say that you plan on moving it to the back yard, but cannot do the necessary yardwork until the spring so asking for an extension to move it.