Your parents will feel that you are abandoning them if you take the dog for a walk or take your daughter to the beach, but your father goes off for a night of drinking with your partner. Why is your father abandoning you like that? Doesn't he want to spend time with you? Why is he spending so much time watching tv? Doesn't he want to chat with you?1?!!?
Just kidding. Of course it's lovely to see your parents but at some point after the first three days of 24/7 interaction, you start to get diminishing returns. 5 weeks of constant togetherness is not going to make up for having been apart for a year and it's pointless to try or to feel guilty about needing to do normal things that don't include your parents. Taking your child to the beach, taking your dog for walks, letting your child watch her favorite shows each day, none of these things will in fact detract from the gift of time together that you are giving to your parents by having them to stay. Can you, at least intellectually, acknowledge this?
Looking at what you've written, I wonder if this feeling of anxiety at any sign of not catering to their every wish, is all an illusion brought on by the fact that your father feels a need to control others and (perhaps unconsciously) uses your willingness to be controlled as a proxy for how much you love him. If you stop letting him dictate what you do all day during "family time" then it's as if he thinks you no longer love him, so he gets into a mood. I know you are putting it down to his age and possibly the beginnings of dementia, but you also say it has always been a bit like that between you two. Maybe this is just how he is. I see that you want to have an optimistic view of his motives, so I won't argue with that, but if you can acknowledge that he likes to be in control then maybe you can also acknowledge that your position as a wife and mother means you cannot always give in, because you have separate obligations to your husband and child. You have to balance your father's wants with their needs. You also have a responsibility to look after your own health because you have to care for your child.
Older women I know who deal with men like this just get into a different frame of mind that works for them to manage this kind of controlling behavior. A sort of briskness that is caring and loving, but no-nonsense. I will be taking Twyla to the beach every day at 5 pm to play in the sand; you're welcome to come, and I'll start dinner when I get back. I'm just going to change the channel to CBeebies so Twyla can watch her programmes and then in an hour or so you can change it back for the sports. I need to take Rover for a walk, who's coming with?
You also need to be somewhat sensitive that elderly people can actually do things that seem strange but that make sense because of their older age. You can talk to your mother quietly to ensure that what you decide upon is manageable for both parents: I can get the sheets off your bed since I know the bending hurts your back but I'll need you to pop them into the laundry while I run to the store. I made beans and rice for us but I made you a nice steak because I know the beans disagree with you.
If this produces doom and gloom or scowling, you just ignore it. Continued cheerfulness. Organise your household in the way that works for you and let them know where they can fit into it.