We are talking about clothes lines in the backyard, right? How could that possibly be anyone else's business? Do they have rules about what flowers you can grow or the material of your garden furniture?
I can understand that some communities might want to make rules about how the front of the house looks like from the street (although I wouldn't want to live in such a place either) but isn't the backyard supposed to be private?
In our neighborhood, we're not allowed to have sheds, and anything on the outside of the house has to be approved by the HOA. That includes any landscaping changes, painting your front door, etc. The back yard is included in that. It's there to prevent egregious abuses, like people raising livestock on their 1/4 acre, and only ever enforced on things like adding a deck or painting the whole house, but at first blush it's kind of ridiculous. FWIW, I've made plenty of landscaping changes (mostly removal, but also planted several fruit trees) without approval, and nobody really cares.
As for why the back yard is included, I don't know for sure, but many of the homes in our neighborhood back onto the retention ponds, so your back yard may be visible from 50 other houses. So if you have 4,000 sq ft of corn in your back yard, it spoils the look of the neighborhood. I'm not saying I totally agree with it, but that's what I believe the thinking is.