How can we say whether the applied emf and the inductor in the the image are in parallel or in series? In some discussions I have seen people taking the applied emf and inductor to be in parallel and concluding that way that the applied emf = back emf, but I believe they are in series and this reasoning will not hold good.
They are both.
They are in series because the current through one must go through the other.
They are in parallel because the potential difference across one is the same as the potential difference across the other.
More importantly, there is a single loop there, and you can use KVL around that loop to show that the potential drop across the inductor must be equal to the potential generated by the source.