The 1880 marriage of Elizabeth Jane Parsons from the 1881 census in Parkstone appears to have been to James Brushette. That Elizabeth was aged 27 in 1881 and born Cerne Abbas. She is consistent throughout the following censuses about her birthplace though her age doesn't quite keep up with the years, possibly because her husband is younger.
On the 1871 census the Parson family is in Wimborne registration district. No obvious death registrations in the following decade or marriages. Nothing is showing for a John and Elizabeth (in any surname for Elizabeth) with a birthplace in Hinton Martell (or just Hinton), or on following censuses with similar ages and a birthplace in the surname Parsons but with just a birthplace of Dorset.
That leaves various possibilities the most common being the army and or emigration for John or a death registration between 1871 and 1881 but somewhere away from his last known residency - the search is obviously not helped by the family's subsequent move to Chorley Lancashire post 1873. Ditto Elizabeth but that could also involve a marriage and a death registration in another surname. Either of them could just be very difficult to find on one census but you wouldn't expect from all three. The difficulty in proving any of these is the commoness of the surname Parsons and the first names John and Elizabeth.
It is always possible that Elizabeth could have chosen to give herself an additional middle name in later life, but that wouldn't directly solve the puzzle of her absences from subsequent censuses unless she used that name as a first name.
She certainly wasn't baptised with, or had her birth registered with a second name.
Regards
Valda