This question contains a linguistic trap. In English (as well as most languages I'm familiar with -- I've heard that there are Bantu languages where this is not true) "exists" means "exists in the present".
Therefore, "does x exist" means "does x exist in the present". The answer to the question "does the past (or the future) exist in the present" would seem to be no.
On the other hand, answers to "did the past exist in the past" and "will the future exist in the future" would seem to be yes and yes, almost by definition.
Thinking about time can be helped by using common sense terminology. I would say that the past, present and future consist of events, not objects. Concerning events, we should ask not whether they exist/existed/will exist, but if they happened/are happening/will happen.
Common sense would say that past events happened in the past, present events are happening now, and future events will happen in the future.
I think that many arguments against the existence of the past and the future are founded on intuitions prompted by misleading terminology.