The problem with talk of "parliamentary" services is that nobody seems to know where to draw the line (no pun intended) of what is/isn't "parliamentary".
A lot of the time it gets muddied with "the minimum required in the franchise" then e heck mom a lot of services are like that - probably most of the services on the ScotRail/ Northern/ Merseyrail/ Wales & Borders franchises (given how loss making they are), probably chuck in all of the Sprinter operated bits of EMR/GWR - I doubt many of those services are profitable enough to be more than the base minimum agreed to in the franchise terms.
Sometimes you see a service diverted an unusual route to maintain route knowledge described as "parliamentary" when it's more about the need to keep up the route knowledge of staff (even if that's inconveniencing the passengers).
So any quirky service that doesn't fit nearly into the hourly service pattern gets called "parliamentary" by at least some enthusiasts - e.g. the York - Pontefract - Sheffield service or the Grimsby - Brigg - Sheffield service - both of which have operated several times a week rather than the "once a week" minimum that seems to be the minimum required to tick a parliamentary box.
There's also the question of how much of a line is the minimum distance to require some "parliamentary" service to keep it open ( e.g. we don't have to run trains into ever single siding on a weekly basis to keep them open, or every loop). There must be some chords on the network that don't see regular action - there are surely loads of bits of "freight" lines that can go over a week without seeing a train running over them, without fear of closure.
As for Lancaster - Morecambe - I don't know why this wasn't replaced by a separate shuttle many years ago - other than some nostalgia for Victorians having a tradition of West Yorkshire workers going on holiday at Morecambe. Chop it at Morecambe, run a much more frequent shuttle service to Morecambe, rather than having the branch on the west coast determined by how many paths down the Aire Valley into Leeds you can find for a Sprinter. Stagecoach run a pretty frequent bus service from Lancaster to Morecambe, the road has always been busy when I've visited, there'd surely be demand for a better shuttle service (even with a fairly basic DMU - doesn't need to be anything more complicated than a 230!).