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There's a nice one in a country park above Linlithgow called Beecraigs which is very good. That would be about 20 mins drive to Grangemouth, mostly down the motorway (my folks used to live in Falkirk).
Beecraigs and others are lovely sites for a break or a holidayhowever, if your working 12 hour+ shifts, your not looking for lovely but cheap and cheerful. Work Eat Sleep. Prices for lovely sites are the same or even more than it would cost for digs here.
There are places where HGV's park overnight, which would be ideal.
Try The Wheel then, near Falkirk. More basic.
I thought being on a site is what OP requested. An HGV layby may not have showers, and may have a 24 hour limit. The council won't take kindly to what looks like a travellers site setting up in a layby.
Fiona no it wont have showers, it is not an HGV layby, that is the reason I have requested the OP mail me lol.
As for the Wheel, most expensive place for what is in fact a car park, not somewhere you would park for 2 nights never mind 1. Might improve in time, but would need to drop the cost by 50% for myself to even consider taking a van there. Sometimes it really pays to know the locality. I know your parents came from Falkirk.