M.P.
I'm a grandparent who has a child friendly house. I'm with you about allowing a toddler/preschooler explore. I also have a friend who was very much into allowing her daughter to explore, touch, pick up things. Her parents were very liberal when it came to allowing their daughter to do nearly every thing. IF she didn't want to get dressed she didn't have to get dressed. That sort of thing. However, they did teach her to not touch things in other people's homes. I went to dinner at an older couples home with them and was surprised to see this 2 yo girl who was active everywhere I'd been with her play with the toys her parents brought and asked before she touched anything.
I suggest that you can teach her to not touch things at Nana's house in 2-3 visits. At 22 months she's capable of understanding the words, "we don't touch/pick up things at Nana's house." Talk about it with her before several times before going to Nana's house. While you're there, remind her. During this first visit prepare to spend your entire time with your daughter reminding her to not touch. Try to do this before she actually touches. When she does begin to play with something, stop her and have her sit down with you for a couple of minutes while you remind her that she is to play with her toys and not Nana's.
My friend walked around the living room with her daughter, holding her hand, so that they could look at all the interesting things. Then the daughter sat down on the floor in the living room while the adults sat and talked.
Be sure to bring several different toys for her to play with. It would help if Nana had a box/basket of toys that she can play with only at Nana's house. If she doesn't have toys, perhaps you could leave the toys you bring so that your daughter has toys that are special because she can only play with them at Nana's.
Teaching her to not touching/playing with things at someone else's house is something that YOU will greatly appreciate as she gets older. Knowing to do this is an important skill for your daughter to learn. It is not too early to teach your child about boundaries. It's one step in learning that everyone has boundaries and everyone's boundaries are different. I suggest finding ways of respecting other people's boundaries is important for you, also. Avoiding going anywhere because adults expect children to respect their boundaries is very restrictive. Wouldn't you like to have a child who knows how to behave no matter where they are? If you wait until she's older you will have a more difficult time teaching her.
I agree that you've made a reasonable request but the answer was no. You cannot change your parents. But you can respect them by teaching your daughter to leave their things alone. Teaching her is also a reasonable request. You can still allow her more freedom at home and in places that are child safe. You are broadening her world and teaching her an important concept and skill that she'll eventually need to learn anyway.