Do models without parameters exist?
Is there any model in machine learning that does not have parameters?


Yes. k-nearest neighbors  is parameterless (there is only a single hyper-parameter $k$).


  If such parameterless models exist, what are their purpose then? Isn't it the whole point of training to tune a model's parameters?


Exactly: such models require no training at all. k-NN in particular relies on knowing the data set upon prediction. Anything close to "training" this model would be pushing data points to a set, but these do not count as parameters.