Re title of the cemetery in question - I stand corrected.
I consulted the Liverpool burial registers (1813-1974) on ancestry which describe the said cemetery as Walton Park Parochial Cemetery. However each page of the said register is printed as follows:
1862-1864 - Free Parochial Cemetery, Rice Lane, Walton, Liverpool
1865 - Parochial Cemetery, Rice Lane, Walton, Liverpool.
I did not check further dates.
Ancestry and/or the custodians of the originals, probably called it Walton Park to differentiate between the other Liverpool cemeteries, eg.
Liverpool St Mary Cemetery
Liverpool St Martin in the Fields Parochial Cemetery.
Yes, Workhouse burials were at all of the abovementioned cemeteries in Liverpool for the various periods.
The registers for 1863 and 1864 for Walton Park cover several volumes, so I would prefer to use the printed title on the register than a Directory entry which I have found are not always accurate.
Hope this helps.