I think this is a tricky one and it all has to do with expectations.
Bottom line: Parents are allowed to ask and have anyone they choose to watch their child and they do not owe anyone explanations on why or why not. Anyone who is asked to babysit by parents is free to say no and not have to give a reason. Both decisions should be respected by others.
All this gets really tricky though because each person has different expectations on how a relationship is going to work.
In my own life:
I prefer to have family members watch my child. I have not found anyone close enough that I would trust to watch our LO, although I would love to find a special person that I love who I can pay. My dear niece who is 13 going on 50, I feel is a great compromise in the future --maybe next year. I would love to pay her to watch LO as he is getting older and will follow direction and DN really wants to watch him. I feel ok asking my DM to watch LO once a month and from her response I think she enjoys it and wants it to continue. I try to follow her lead. My ILs probably watch LO twice a month if needed, sometimes my DM is out of town. I actually think they would prefer more time but I miss my LO too much to give up too much time unless I have to, as some months are busier than others. For example November and December were really busy, but I only needed ILs to watch LO once in January (Birthday party) and so far it is only looking like once in February (I had the stomach flu).
My DH feels that grandparents should be ready at any time to watch the grandchild. This is one of those frustrating parts of our marriage and in the grand scheme of life not a big deal but still annoys me. I am not sure where he gets this mentality from because his family lived 10 hours away from his grandparents, although they visited a lot. I am the one who lived with my grandmothers. For example, my DH was in charge of LO two weekends ago. I suffer from really bad morning sickness (yay! one of the lucky ones), almost debilitating at times. Also my LO brought home a horrible stomach virus that weekend which I caught. It was so horrible, I took care of LO two days, then I was sick one and then LO started coming down with a second virus before I was done and morning sickness was horrible. It took me 5 days to eat food and I still hate eating. DH took LO to his parents Saturday morning, brought him home for his nap, and took him back to his parent for the afternoon. Then on Sunday, took LO to church (this was before LO showed sickness and it had been over 24 hours since his first bug), brought him home for his nap and then took him to my mom's house for the afternoon. I think he would do this every weekend if he was left to care for LO by himself.
I just think its up to each party to find a happy balance (if you care to make the other party happy) on each ones expectations. I think the real trouble is when expectations are so extremely different.