There are apps monitoring child's movement based on their phone location (e.g., MM Guardian). These apps can also monitor children's use of their smartphone and include applying restrictions for certain features. None of these have been investigated so far unfortunately and we lack even preliminary data on how effective their are in helping parents and their impact on parent-child relationship.
There's this app that parents can use to lock the phones of their children unless they respond. I think it's a slippery slope and can be misused in many ways, creating enormous stress on children and on parent-child relationships.
There is a lot of research about parenting the internet and via social media, such as this one (which uses the term "technoparenting"):
Yardi, S., & Bruckman, A. (2011, May). Social and technical challenges in parenting teens' social media use. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 3237-3246). ACM.
... but I haven't come across any specifically about mobile APPs.