Is their a legal definition of what constitutes a parliamentary service and what is just sparse running?
No, not really. I think probably we can differentiate between stations that have a daily service in both directions and those which are less frequent. A daily service makes a station usable for the general public, particularly if it's a morning commuter and evening return service.
An example of the latter not already mentioned on this thread is Pegswood, Widdrington, Acklington and Chathill in Northumberland which each have a morning and evening southbound train to Newcastle, and one evening return train from Newcastle per day (though Chathill has an extra arrival from the south in the very early morning, which is really just an alternative to an ECS running). While minimal, this is clearly some level of usable service and different from a parliamentary.
Leaving stations open with a minimal service also makes it much easier to introduce a more comprehensive service at a later point: this has happened in Dunston for example, and I think is on the cards for the Northumberland stations as well at some point?