This is a historic British railway term describing a class of service mandated by law to provide affordable transport. Railways were required to run at least one such service daily in each direction at a fixed low fare. In later usage, the adjective describes the bare-minimum service run solely to avoid the legal and administrative burden of officially closing a route.
They only run a parliamentary service out there on Sundays to keep the route officially active.
North American railroaders rarely use this term, as the regulatory framework for route closures developed differently under the ICC and later Amtrak.
Related terms
Parliamentary train
A passenger train run at the absolute minimum frequency required to keep a route legally open.
Passenger train
A scheduled train consisting of equipment designed to transport people.
Local
A freight train that works the industries along a line, setting out and picking up cars at each customer.