Parry was commissioned lieutenant on 20 August 1759.
On 11 January 1775 he was promoted commander and appointed to the Cruizer 8, in which he served for a long time off North Carolina and Cape Fear before she was deemed unfit for service and destroyed by order of Vice-Admiral Lord Howe on 2 October 1776. From the spring of 1777 he commanded the new sloop Lynx 16, operating out of Spithead and cruising between Portland and Rame Head in search of American privateers. In the autumn he took a convoy out to the Leeward Islands in company with the mutinous sloop Shark 16, Captain John Chapman, most of whose standing officers were confined for inciting the disaffected men.
After returning home Parry was posted captain on 7 April 1778, and appointed to the Europa 64 with the flag of Vice-Admiral John Montagu, going out to Newfoundland in May after the King had reviewed the fleet at Spithead, and relinquishing the command after returning home in the autumn. On 7 January 1779 he sat on the court martial of Admiral Hon. Augustus Keppel following the Battle of Ushant on 27 July 1778, and that winter he held the acting command of the Proserpine 28 in Vice-Admiral Lord Shuldham’s squadron, operating out of Plymouth.
During 1779 he joined the Lizard 28, recapturing the cutter Jackal on 18 May 1780 in company with the cutter Busy 10, Lieutenant James Cotes and serving off the Orkneys. In November 1780 he recommissioned the Actaeon 44, going out to Jamaica in the following March and returning in 1783 to be paid off.
In 1787 Parry commissioned the new Adventure 44, serving as commodore off Guinea, before being invalided home.
After remaining unemployed through the Spanish Armament of 1790 he commissioned the Prince 98, serving aboard her from 2 October 1794 to 9 June 1795, and in the spring of the latter year he also sat on the court martial of Captain Anthony James Pye Molloy who was found guilty of misconduct at the Battle of the Glorious first of June.
Parry remained unemployed following his elevation to flag rank on 1 June 1795, but was promoted vice-admiral on 14 February 1799 and died at Hythe near Southampton on 18 December 1803 after a long illness.
On 27 July 1771 he married Fanny Eames, niece of the M.P for Newport Isle of Wight, and they had issue at least four daughters.