Homeowners Associations are tyrannical and usually intolerant about difference between yards. They are put in place by the builders because they can collect fees and use that money to put in landscaping, walls, perhaps a public pool area, and once the neighborhood is built out 80% they turn management over to a for-profit company in the business of running homeowners associations. The fear is that if someone's house or yard has a unique appearance, their home values might be downwardly affected. Personally I would never live somewhere that has a mandatory HA in the builder's covenants. /rant off
I've considered writing a book, with lots of photos of subversive gardening that can be done in neighborhood held hostage by homeowners association.
Grass doesn't grow well in deep shade; the HA probably wants the trees to stay in place, so they have to let you choose - either cut down the trees so you can grow grass, or let you plant something that will survive in the shady area. Can you mix clover in with whatever is growing there now? It's healthier for the grass that is there to have the nitrogen-fixing clover in the mix, and clover can be mowed just fine. Check to see if any of them are more shade tolerant and keep it a low-growing variety.
I'm afraid this isn't much help, I really detest homeowners associations. I'd be out there butting heads with them. (When my children were small one of our favorite books was Daniel Pinkwater's
The Big Orange Splot)