Is parsing JSON on the serverside dangerous?
Well... to take any user input is potentially dangerous, especially when using external libraries you don't know well. For instance, it's simple to demonstrate how a quicksort algorithm poorly implemented can be used to a Denial of Service attack by feeding it tailored data that leads all iterations to an O(n²). Does that means sort algorithms are dangerous? No. It means poor implementations of anything can be dangerous.
Obviously, you should estimate the danger and benefits and choose what to adopt based on that. Frankly, if you were working on the kind of project where the potential risk of using a 3rd party json implementation is a security issue, you wouldn't be asking about it on Stack Overflow.
I'm writing a Python application with Django and serializing objects to and from JSON using Python's standard json library.
I've heard something along the lines that 

any seralizer/deserializer/parser is a potential security threat to the application

Is this really true on the serverside? Do I need to fear someone sending tampered JSON up to my instance and them having remote code execution privileges? I would think that there wouldn't be any security threats other than (potentially) XSS or SQL injection, which in my case (converting values to ints, storing all data in MongoDB) shouldn't be the case.
Should I be paranoid along these lines of parsing JSON data from clients?