Whitefriars Road is a part of “the Friary” area, which is named after a Friary that was on the site. The “White friars” were the Carmelites. According to A brief history of salisbury, there were both Franciscan (Grey Friars) and Dominican (Black Friars) were active in Salisbury, but I’ve not yet found any reference to the Carmelites.
[...] Whitefriars Road – the ‘White Friars’ were and are the Carmelites. The Carmelites were a group of monks that formed on Mont Carmel in Isreal in or before the 13th Century. They arrived in England in 1241-2. The monks were known as ‘White Friars’ because they wore white habits [...]