Sadly I think this kind of friction is common when a new mother doesn't parent the same way that they were parented.
My mum is a mixture of supportive and unsupportive elements. Recently I noticed that the supportive bits were to do with the parts of parenting she did with me (breastfeeding until 6 months, cloth nappies, responsive to baby, moses basket in our room)
But elements of our parenting choices which opposed her own choice are always critised (extended breastfeeding, ECing, babywearing, co-sleeping)
Now I put forward factual reports on the safety elements of these things and with future comments I just explain that 'this works for our baby and for our family'
It still hurts when she disagrees with our choices, but she respects us enough that she will potty my DD and feeds her expressed breastmilk when she babysits.
I think my mum realised this recently when she quiped to my father that she parented "all wrong" according to todays studies. This did upset me and I explained that there are many ways to parent and I need to follow the recent guidelines which I have access to today, just as she followed the health guidance she was given.
I don't want her to think that because we parent differently that I am critising her parenting, on the contrary I had a wonderful childhood and often remind her of how happy I was growing up in our family (outside school)
I had the slings are dangerous argument but better that she is strapped to me and I have BOTH arms are free to break any fall, than a mum pushing an empty pram and holding an infant on their hip with their other arm. Also what happens to babies in prams if the mother fall, I would think some prams would be left in or roll into a dangerous situation. i.e. There is no absolutely safe way of transporting a baby anywhere.