Can every program be parallelized infinitely and automatically?
If you're working in the strict lambda calculus, everything can be automatically parallelized. In particular, when evaluating a function application, the function and the argument can always be evaluated in parallel.

However, it cannot be infinitely parallelized. There are inherent data dependencies: the result of a function application can't be determined until both the argument and the function have been evaluated, meaning that you need to wait for your threads to both finish, then synchronize.

This is still relevant with your clarified definition of infinitely. For example, if you have 5 processors, it's possible that a particular program can only ever use 4 processors, because of the data dependencies.

Moreover, while this is automatic, it is not "performance for free." In practice, there is non-trivial overhead to creating and synchronizing threads. Moreover, it's difficult to do this in a way that scales only to the current number of processors: if you have 5 cores, the automatic parallelization might generate 6 threads, and in general, it's not possible to know at compile-time how many threads will be active at a given time. 

So, you can automatically make a program that runs massively parallel, but with the current state of affairs, it will likely be slower than your original.

It's also worth mentioning that, in practice, this becomes difficult with shared access to resources and IO. For example, a real world program might write to a disk, which can cause problems if done in parallel without control. This is something that can't be modeled by the standard lambda calculus.