I suggest that you are being overly sensitive in part because you don't know these neighbors. It's easy to focus our daily frustrations on someone we don't know because we don't have our view of them balanced by being able to think of them as people just like us.
I suggest that you take the time to get to know them before trying to talk about what they're doing. If you already say hi when you see them outside, take the time to stop and ask how their day is going. Make any positive comments that you can such as your yard really looks nice. I see you working to make it nice. If their kids are close in age to your kids, perhaps suggest a play date with them in your back yard. State specific times to start and finish unless you're comfortable with your kids and their kids hanging out together.
I get irritated when someone parks in "my" space on the street but then I remind myself that it is a public street and find somewhere else to park. My irritation goes away. Eventually, I was able to then park in "someone else's" spot when no other was available and I had things to take into the house. Otherwise, I've sometimes parked down the block, reminding myself that walking is good for me. In other words, I deal with my feelings so that I can stop projecting my frustration onto the neighbor.
I also suggest that your neighbor doesn't know that these things are upsetting you and unless you talk with them in a calm, non-judgmental and friendly way you can't "blame" them. Your judgment of what they're doing is what is causing your frustration and irritation. I suggest working on looking at what other people are doing as that is just the way it is without labeling them inconsiderate.
It also helps to try to put yourself in their shoes so that you can get closer to understanding why they're doing it. From your description it seems to me that they have good reasons to do what they're doing. Mowing early in summer's hot weather makes sense to me. Not parking in front of his yard makes sense. If the car were parked in front of where he's mowing his car may get splattered with grass or he can't edge on the street side.
My neighbor sometimes mows a strip into our yard and sometimes we do the same. It's difficult to see where the property line lies. And our houses are close together. They mow all of the grass in between our two houses. When our "flower bed" on that side of the house was over run with weed they talked with me about it. I hired a boy to clean out the beds.
Being in a neighborhood requires co-operation and compromise. When a person make judgments based on their needs without consideration for the neighbor's needs they are setting themselves up for frustration that leads to anger. Nothing gets resolved. I suggest you're in that space now and if you can find your way out to focusing on how you can live peaceably with your neighbor while taking care of your needs and their needs thru conversation and compromise you'll be much happier. The blame game never works!
I've lived in my neighborhood for 30 years. Most of my neighbors have lived here for many years but yet I don't know them really well nor do I socialize with them. When my daughter was a kid, she did play with neighborhood kids and I found this very helpful. I did get to know the mothers better and we ended up watching each others kids while one of us ran an errand or wanted an hour without kids. Is there a reason you are keeping your kids away from their kids? They sound like responsible parents.
As to the father never asking if it was OK, I suggest he is thinking that you would tell him if it's not. That is what I expect. I think it's counterproductive to relaxed living to always be concerned if I or my child is offending someone. I expect them to tell me if we are. If they give me "dirty" looks and the situation is working for me, I sigh and say, that's their problem and continue what I'm doing.
We are not on this planet to make other people happy. We are only responsible for our own happiness. This responsibility does require that we be honest with other people in a kind way. Compromise and good feelings are the focus in my day. When I'm friends with someone I'm more focused on being a part of their happiness than I am with a total stranger.
Later: After reading more of your responses, I was reminded that many years ago, I worked graveyard and in the summer the people in a house behind mine had kids who played in the back yard. I walked around the block and told them I worked graveyard, usually didn't' get to bed until around noon and could their children play outside in the morning instead of the afternoon. I suggested that if they weren't yelling playing outside wasn't a problem. It was just the extra loud noises that were intermittent that woke me. No problem! The children continued to play outside on some afternoons but they were much quieter and it was OK. I didn't know these people at all. I was diplomatic and so were they.