Should our tiny team use Parse or firebase or other backend solution for a global instagram-like app we are developing ? Speed of development is very key.
If you are looking for speed of development, the best database back-end is the one that your team has the most experience with that fits your needs.
I have experience with relational databases, so I would likely start with a relational database like MySQL. If I were more familiar with key/value stores I might say something like Riak. I think your best bet is to go with what you are familiar with to get a prototype of your app off the ground, then work on finding the best tool for the job after that. Make a list of the features you need and find the back-end that best supports those features, and has the most active community.
I think a lot of people get so hung up on implementation details when developing new apps that they never make any progress on the actual app, they instead put all of their effort into learning some new technology. Refactoring is hard, but not as hard as learning a completely new technology while still trying to iron out general design/architectural issues.