In 1803 Palgrave was articled to a firm of solicitors, but was called to the bar at the Middle Temple in 1827.
His father, Francisco Antonio Zumalacarregui, was a lawyer who possessed some property, and the son was articled to a solicitor.
Having passed his law preliminary, he was articled to a firm in Portmadoc, and in 1884 obtained his final qualifications.
William was educated at Oakham and Newark grammar schools, and in 1714 he was articled to Mr Kirke, attorney at East Markham, in Nottinghamshire.
He was articled as a law clerk in Edinburgh, and his Elegy on Craigmillar Castle (1776) was printed during his clerkship. In 1781 he removed to London to devote himself to literary work, publishing in the same year a volume of Rimes of no great merit, and Scottish Tragic Ballads.