Born in Whitby on 7 February 1834, he was the younger child and only son of William Pannett (d.1July 1858), who had come to work in Whitby from Littlehampton in Sussex and who soon after marrying Alice Elliott became a hairdresser. At the age of 12 Robert was apprenticed and then articled to Thomas Broderick Simpson, a solicitor, after having attended a private school run by Ralph Gibson.
On Simpson's retirement Pannett was assigned to Bell, Broderick & Gray, qualified at Lincoln's Inn to be admitted as a solicitor in November 1858, becoming a partner of Gray in Whitby. He became very active in public affairs, first on the Town Improvement Commission from1863 (being its clerk from 1866), then from 1872 a member of its successor, the District Local Board, until Whitby became an Urban District Council in 1894. On the North Riding County Council coming into being in 1888, he was elected as a Liberal for Whitby West Ward until 1908 when he was elected to the aldermanic bench, being Chairman of the Technical Instruction Committee. On the opening of the County Hall at Northallerton in 1906 he established a library and became its librarian. Other offices held were as a magistrate from 1903, Chairman of the Whitby Water Company from 1899 to 1915, and a director of the Whitby & North Yorkshire Building Society.
Pannett, who never married, was a generous benefactor to the town and supported numerous charitable causes, including the cottage hospital, the Trust for the Blind, the Charity Organisation Society and the Choral Society. From 1861 he was a member of the Literary and Philosophical Society, serving on its committee, and in his will left money to purchase land for a public park, where the Pannett Art Gallery and Museum was opened in 1928. A member of Brunswick WM chapel, he organized its Sunday School library, gave the 'Brunswick Room' as a school hall in 1892 and left money for the installation of a pipe organ.
Pannett retired from his legal practice in 1896 and died on 22 July 1920,