Do other universes like ours exist? If they exist, how do we know that they exist when we have even not seen the ends of our own universe?
The simple answer is (as with so much in astronomy):
Parallel universes may or may not exist. There is no definitive way to prove that these universes do or don't exist.
A parallel universe is a separate existence to ours. The Theories that suggest that there may be parallel universes are classified as theories of multiverses. There are many theories of multiverse, all of which propose different ideas about what could exist beyond the limits of our universe. There are also theories that suggest that the multiverse doesn't exist, although the theories with most support are by far the multiverse theories.
For a nice reference in book form, see Steinhardt and Turok's "Endless Universe: Beyond the Big Bang". Also, see Max Tegmark's work on multiverses levels I-IV (Max Tegmark -> See his Scientific American article entitled Parallel Universes).
If they exist, they are out of anything we can ever reach. They are (by definition) out of our Universe, so we can not affect them and they can not affect us in any way. That's the same as saying that, for any experimental definition, they do not exist.