No, when it's in session Parliament is pretty much a full day's job for an MP, probably more if you're actually trying to do good work and/or have a cabinet or shadow cabinet post. Remember in the UK the executive is run by ordinary people elected to the legislature, they aren't just arbitrarily picked by a President, although in some roles it will be tolerated to "elevate" outsiders to the House of Lords and then give them a minor cabinet post, doing this with a Great Office would cause outrage in the modern era, those are always given to elected MPs. Because the Prime Minister can change in days (or in practice hours) an entire parallel Shadow cabinet also needs to exist for a party that would seek government, with no power today but every capability to take over political control of government tomorrow if called to.
My local MP commutes, less than two hours each way - gets the same train home I was catching after visiting the (London) office, so leaving London just after 1900 on a weekday, to obviously the same city I live in on the coast.
Those who live further away have flats in London, they stay for four nights a week and go home at weekends. Indeed some of the last train services heading North on a Friday are almost explicitly for such MPs and staff. Kings Cross to Edinburgh on a Friday evening is not exactly a tourist route, it won't get there until after midnight.
Using an official car would be an unacceptable expense except for certain senior ministers (who work in London all the time, e.g. the Prime Minister and often senior cabinet ministers live there) and it's frowned on even then, one of the few good things you can say about Boris (Johnson, the city's previous mayor) is that he rode a bicycle a lot rather than constantly getting official cars just a few miles up the road.