The term comes from the minimum level of service specified by parliament (a long time ago) for a line (or station) to be considered 'open' and avoid closure proceedings; see
https://www.railforums.co.uk/threads/trivia-what-is-the-definition-of-a-parliamentary-train.129742/
The minimum frequency for a route to be considered to be open for passenger use is only once per week (in one direction!)
Very few trains run
that infrequently!
However there are many more examples of
infrequent services, which run more frequently than the parliamentary minimum, but only really exist for route knowledge retention purposes, e.g. Paddington to Marylebone, which runs once per day.
Then there more still that do not just exist for parliamentary or route retention reasons, but to provide a service at a time that is genuinely useful to customers e.g. the Dundee to Glasgow train via the Forth Bridge.
Then there are services which use lines that each have frequent services but the
overall route taken by that train is infrequent. But this is getting even further than the original request.
If you want a narrow definition of strictly parliamentary trains it will be a very short list (with possibly none in the South West of England) so you may wish to widen your requirements to one of the above?
In any case you can travel on any service advertised in the public timetable.
