Why would you use a subset of the training set as the validation set in parametric classification? I was told by my ML lecturer that the validation set, as used in parametric classification, is used to determine how overfitted the model is to the training data, but that it is also a subset of the training data. I cannot understand how this would make sense since it seems intuitive that if the model is overfitted to the training data then it would also be overfitted to the validation set.

I suspect this is just a mis-communication due to an ambiguity in the language.  Often the entire set of data available for the ML process is referred to as "the training data".  However this data is then split into the training and validation sets.  So I could imagine situations where one said "training data" could intend either a) the entire ensemble of data available, or b) just the set of data actually used during the training process.
For what it's worth, I find "training [data] set" and "validation [data] set" relatively unambiguous ways to refer to the set of data used in the training/validation process.
"Training data" can be used to refer to entire ensemble of data available for ML.  At the very minimum it is ambiguous, and possibly in some circles* a downright error to use the term "training data" to refer to the set of data actually used during the training process.
The training does not utilize the validation data.
[*] I'm not aware of anyone who's tried to set this as "a rule" but it might be a way to cut the ambiguity.
