There's a huge park in Allestree and it isn't too far from Markeaton Park and there's loads to do in there (paint a pot, ducks, swings, miniature golf, paddling pool in the summer, farm, cafe's, stream train etc.) I think there's a play group in St John's church but I'll have to check on this. Ds and I have to go to Littleover on Monday so we'll see if there are any notices and get back to you.
They do hold an after school club there which collects from Lawn, Portway and Markeaton (Kedleston Road). Ds went to Markeaton Primary for a term and a half before I moved him into a private school (we were not impressed, put it that way).
The city centre is accessible from both Littleover and Allestree but the traffic from Littleover to the city centre is an absolute nightmare. There's now a bus lane on Kedleston Road (Allestree) which helps. There are parks in Littleover but not as nice as in Allestree (I must sound as though I'm biased, I'm really not and I don't live there). There's a row of shops in Littleover (Iceland, Post office, chip shop, take away, co-op, Barclays, a pub, wedding dress shop etc) and a small shopping arcade in Allestree (Co-op, post office, flower shop, cafe, charity shop, Natwest, health food shop, Farmfoods etc)