Production of the Sunbeam Alpine between 1959 and 1962 was contracted out to Armstrong-Siddeley at Parkside. To the best of my knowledge, Rootes had no financial stake of any significance in Armstrong-Siddeley. This was simply a convenient business arrangement.
According to Coventry Car Makers, Parkside was demolished in 1996.
It never occurred to me at the time of my visit to Coventry to try to find the site. I believe it was located on the block formed by Park Side, Puma Road and Mile Lane, just off the Coventry ring road. In the A-Z Street Atlas of Coventry, see Map 3, G6.
If you have Google Earth, you can see recent satellite images of this area by opening this .kmz placemark file: