Parkinson always called them Vises - I saw a comment that Vice would have offended Victorian sensibilities; I've no idea if this is correct!
Parkinson certainly survived longer than the 1930's - their vices are featured in my 1964 Buck & Hickman catalogue, but not the 1971 version. Whether because they had closed, or because B&H had stopped selling their products, I'm not sure. They were part of a company who also sold grinding machine tools, I believe under the name Parkson.
As has been said, I think they invented the, now common, quick release mechanism - these feature in the 1913 B&H catalogue. For woodworking vices, I think the design of the front plate changed over time in the logo etc that was cast into it.
My main interest is in the engineering vices where they made 3 main ranges - the names may have varied over time :
Handy - aimed at the home user, cast iron, plain thread, ie no quick release. Sizes 0A 2.25/2.00 width/opening to 3E 8.00/9.25
Perfect - mechanics, cast iron, quick release. Sizes 5 3.25/4.00 to 9a 7.50/10.00
Samsonia - engineering, cast steel, quick release. Sizes 35 3.25/5.00 to 39 7.00/8,.00
They also made a couple of "funnies" :
Ball base vise - an elegant design of small vice mounted on a ball with a cam to lock it in position. Only one size, I believe A 4.00/3.00. I'm currently restoring one.
Swivel jaw - where the rear jaw could swivel about a post, or be locked parallel with a pin. I found one recently which is awaiting attention. 3 sizes from 7 4.25/5.00 to 8a 6.00/8.00
Record eventually dominated the vice market with a staggering range of vice designs and sizes - probably far too many variations with huge working capital / stocking costs. When they were eventally sold to Irwin & production shifted to China, the range contracted a lot - to the extent that you wonder why Irwin bought them, except for the name.
When people look for a vice on Ebay etc, much of the focus is on Record - but vices of at least the same quality are available from Parkinson, Fortis, Woden, Paramo.