Parry was commissioned lieutenant on 20 August 1759.
On 11 January 1775 he was promoted commander and appointed to the sloop 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.
On 10 March 1777 he was appointed to the new sloop Lynx 16, which was fitting out at Deptford. After arriving at Portsmouth from the Downs on 7 June she came into Cowes on the 20th following an unsuccessful cruise around the Isle of Wight in pursuit of an American privateer. Thereafter operating out of Spithead, she also cruised between Portland and Rame Head in search of enemy privateers. On 27 July she arrived at Cork, and on 3 September sailed for Jamaica in the company of twenty-nine merchantmen, although within three days she was obliged to return with her charges because of adverse winds before putting out again later in the month. Her consort in the voyage out to the Caribbean was the mutinous sloop Shark 16, Captain John Chapman, four of whose standing officers had been confined by the time of her arrival in the West Indies for inciting her disaffected men.
After returning home with the Lynx in the spring of 1778, Parry was posted captain with seniority from 7 April and appointed to the Europe 64 with the flag of Vice-Admiral John Montagu, which vessel attended the King’s review of the fleet at Spithead in May. He was succeeded in this command shortly afterwards by Captain Thomas Davy.
On 7 January 1779 he sat on the court martial of Admiral Hon. Augustus Keppel which investigated that officer’s conduct at the Battle of Ushant on 27 July 1778, and in March he was appointed to the Lizard 28 at Chatham. In the following month newspaper reports revealed that his frigate’s sailing master had been shot and killed when attempting to press seamen from a collier at Rochester. The Lizard was attached to the Grand Fleet under Admiral Sir Charles Hardy in the summer, and after entering Plymouth on 4 July she was sent to patrol off The Lizard where she discovered the allied fleet on its voyage towards the Channel. In the autumn she cruised in the North Sea where in one particularly heavy storm lasting two days a man was drowned after falling from aloft, whilst the hands were kept constantly at the pumps. The Lizard returned to the Downs in November, and once back at sea she recaptured the cutter Jackal on 18 May 1780 in company with the cutter Busy 10, Lieutenant James Cotes. She later underwent repairs and copper-sheathing at Chatham before leaving that port in early June, and not long afterwards she was sent with a squadron under the orders of Commodore Hon. George Murray of the Cleopatra 32 to the Shetlands to protect the homeward-bound Greenland whalers, whilst in November she escorted a number of Hudson Bay whalers to London from the Orkneys.
In November 1780 Parry recommissioned the Actaeon 44, which took aboard bullion at the Nore in February 1781 before going out to the West Indies from Portsmouth with a convoy in March. Remaining in the West Indies, on 5 June 1782 she came upon a three hundred strong Franco / Spanish convoy in the Windward Passage under the escort of twenty or so sail of the line, and on the following morning she managed to capture the stern-most merchantman before being forced to flee from the men-of-war. During the late summer Parry commanded a small squadron in support of a land force that defeated a Spanish attack on the Mosquito Coast settlements, prior to returning to Jamaica in October, and the Actaeon eventually came home in 1783 to be paid off in July.
In January 1787 Parry commissioned the new Adventure 44, arriving at Portsmouth from the Downs on 31 March with orders to serve as the commodore off Guinea, and to deliver troops to reinforce the garrisons on that coast. By October his command was back in the Downs, and she sailed once more for Africa from Portsmouth in early January 1788 before returning on 24 May. In June Parry attended a meeting with prime minister William Pitt, together with a number of slave trade delegates from London and Liverpool, as well as the influential Comptroller of the Navy, Rear-Admiral Sir Charles Middleton, to discuss a forthcoming parliamentary bill aimed at correcting abuses in that trade. In October it was announced that he would not be returning to Africa, apparently on account of his health, and he was succeeded aboard the Adventure by Captain John Inglefield.
After remaining unemployed through the Spanish Armament of 1790 and the early stages of the French Revolutionary War, Parry commissioned the Prince 98 for the Channel Fleet at the beginning of October 1794. In January 1795 his command was fitting out at Portsmouth where her crew was completed by a transfer of all the men from the Canada 74, Captain Charles Powell Hamilton, and in the first two weeks of May he 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 did not see any further service following his elevation to flag rank on 1 June 1795. He 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 at Alverstoke in Hampshire he married Fanny Eames, the niece of John Eames, the M.P for Newport Isle of Wight, and they had issue at least four daughters.