Answers from a BUSLIB listserv question:
You can consult publications from the NCES (National Center for Education Statistics, such as the annual (see also earlier years): Parent and family involvement in education, from the National Household Educations Surveys Program of 2012 (NCES 2013-028) - (http://nces.ed.gov/pubs2013/2013028.pdf)
There is, a new book from Harvard University Press on this topic: Robinson, Keith and Angel Harris. 2014. The Broken Compass: Parental Involvement with Children's Education. Cambridge, MA: Harvard University Press (http://crsi.nd.edu/evaluations/new-research-on-parental-involvement-in-schools/).
Parent and Family Engagement, from the U. S. Dept. of Education (http://www.ed.gov/parent-and-family-engagement)
Child Trends: http://www.childtrends.org/?indicators=parental-involvement-in-schools#sthash.s0sLCfPN.dpuf
The Center for Public Education has: Back to school: How parental involvement affects student achievement (full report) - (http://www.centerforpubliceducation.org/Main-Menu/Public-education/Parent-Involvement/Parent-Involvement.html)
There's also the National Educational Association - a search of their site yielded 238 hits (for parental involvement and after school) - (http://www.nea.org/home/37004.htm?q=parental%20involvement%20and%20after%20school )