Parsonage Green in is Harnham, Parsonage Mead is in Wilton and Parsonage Close is in Stratford-sub-Castle. These are to the South, West and North of Salisbury respectively.
The word ‘parson’ itself seems to have the same root as ‘person’. The Online Etymological Dictionary suggests it may be an abbreviation for ‘persona ecclesiae‘, meaning a ‘person of the church’ [1]
I don’t know whether all or any of these roads are named after actual Parsonages (a parsonage being the house of a parson). They are all fairly close to churches, but most roads in the area aren’t too far from a church. If you know, please leave a comment.