Snowchym, I think that you have an idea of what your father and mother should be doing, but it doesn't fit with reality. You also get mad instead of trying to resolve the issue. You've mentioned that before, when you refused to allow your father to use the bathroom.
I understand completely. I had the same thoughts and feelings about my ex-husband. I should not have had to ask him to do chores. If he did do a chore, why should I have to thank him? He never thanked me for doing the chores I did. I often expected him to be a mind-reader. OF COURSE I wanted him to take his lunch kit out of his car and at least rinse it. OF COURSE I expected him to drive our daughter to the doctor when I broke my arm and could not drive. OF COURSE I expected him to understand why we had to pay for taxi rides to and from the doctor's office. (My daughter was sick and taking the bus was well over an hour trip both ways.)
Needless to say, I had expectations and ideas and he did not have the same expectations and ideas. Was it all his fault? As much as I want to say "yes", it's not true.
It would have done me no harm at all to communicate. "Would you please soak your lunch kit when you get home?" Or, "I cannot drive and our daughter needs to see the doctor. Would you prefer us to take a taxi or could you drive her?" My not asking was a form of passive aggression. I was setting him up to fail. Oh he still might have failed -- anyone could. He would say, "I mowed the lawn." I could have easily said, "Thank you!" and then told him, "I cleaned the bathroom." This is a personality difference. I did not feel the need to say I did a chore and wanted praise, but he did. How much harm would it do me to to simply notice or acknowledge him? None.
So (finally I get to the point) I think you should either invite your parents to your place or offer to bring a sweet treat over to their place and say you'd like to sit down for a chat.
Start by telling them that you love them and you think that you've all been having a difficult time lately with your relationship. You need to tell your father that you made a mistake. You should have moved the bed, but you did not think that he would mind helping you move the bed. So agree that in the future BOTH of you will try to say what you want and expect from each other. He expected you to 'know better' and you expected him to 'just know you wanted help with the old bed'. Neither of you was right -- or wrong. You miscommunicated. If you do not clear it up, then it adds another layer of misunderstanding to the on-going problem.
I also think that everyone in your family seems to have a 'right or wrong'/ 'do or die'/ 'my way or the highway' attitude. Life is not like that. It is anything but black and white. The shades of grey way outnumber the one way or another premise. So try to communicate. Remember that these are your parents. You could not be the great person you are without them.
When and if your parent makes a mistake in the future, or tells you that you made one -- just take a minute. "Wait, I think we're misunderstanding each other. What are you thinking? This is what I am thinking." Sometimes you will have to allow them to 'win', even if you disagree, If anyone is trying their best to help you -- that's a good time to let them win. Parents have certain 'rights', too. They are owed your respect. That doesn't mean they don't need to respect you, but kids earn respect and parents by virtue of being your parents already have respect. (I am not talking about specifics here, just in general about respect.)
On Edit: You father should not swear at you. You should be polite to your parents. We were not there, Snowchym. We can't understand why he swore. Exasperation? He always swears? He's nasty? He can't seem to make you understand what he may think is obvious?
You cannot make people honestly apologise. Anyone. Ever. They have to understand what they did was wrong or hurtful and then they decide whether or not they apologise.
You earn respect. Please try to sit down and talk it out with your family. I don't know how it is in your specific culture, but I think in general most parents are respected and shown respect simply because they are older than you and because they are your parents. You need to give them reasons and show them that you deserve respect.
I do admire that you are trying to get it right. I suspect that in time, you will have a wonderful relationship with your family. Best of luck.