As this was recently active I'll add my 2€
I've looked around for a few months and the best one I have found so far is Loopback which is a MEAN based service and its open sourced. It seems to have a few heavyweight users including some who explicitly say they have used it as a migration route from Parse.com
http://loopback.io/
Technically it has a lot of similarities to the Parse stack with support for User accounts etc. Its opensource and I have so far brought it up on AWS for which the creators provide a preconfigured AMI
This does not have exactly all the same parts as Parse, but the core ones are there and you can roll whatever you like.
I intend building a front end to Parse using this and express to fill areas that Parse doesn't do and to provide a means of migrating from Parse that retains broadly the same technical/architectural tools. The associated Strongloop tools/service appears to provide a nice ops package once one leaves the managed service provided by Parse.
One big advantage is that you are not restricted to any particular database and if the app would work better with MySQL you can use that or something else instead of Mongo.
I have yet to find anyone else who has used it to compare notes.
Edit :
Loopback (or the company) has now been acquired by IBM, from my perspective this means it better competes with Parse on a strategic level as it has a major backer for the technology. I'm still using Parse on my own (and customers) projects but looking to try Loopback on a future personal project.