Parry did not participate in searches for Franklin, though he probably was consulted during the last ten years of his life, as Parry had considerable knowledge of the Arctic Archipelago.

William Edward Parry was born to a well-known family at Bath, that retreat of the influential. At the age of 13, he joined the navy. Three years later, in 1806, Parry was designated a midshipman; in 1810, a lieutenant. In these years Parry got his taste of the frigid north when on guard duty to the Spitsbergen whale fishery (Spitsbergen being between Greenland and Norway). In 1816, he wrote "a small volume on Nautical Astronomy by Night. He was at the North American station during the years, 1813–17.1

In 1818, Parry was with John Ross in the ship, Alexander; they were on an arctic expedition. While the Alexander did follow a course that included the coast of Baffin Bay, no new discoveries were made; they apparently were back in England before the year was out. In 1819, with the Hecla and the Griper, Parry returned to Baffin Bay. The pair entered Lancaster Sound and sailed "right through the mountains, John Ross had suggested enclosed the inlet. Prince Regent Inlet was investigated (and named) but found to be blocked by ice."2 But the west continued to show promise, so they continued west into Melville Sound. (See Map.) The winter, however, as I am sure was expected, overtook our heroes. They found shelter on the south coast of Melville Island which they named, "Winter Harbour." A long winter, which for Parry and his men, extended to ten months; but they got out of it and sailed for England. The importance of Parry's Expedition of 1818-9, is, that "Parry had demonstrated that with sufficient provisions, a ship and crew could winter successfully above the Arctic Circle."3

www.rmg.co.uk did such a good job of this, I quote:

"Unsurprisingly, Parry was asked to return to the Arctic. Remembering the pack ice that had halted his progress within sight of Banks Island he suggested searching for a more southerly route through the north-west corner of Hudson Bay and set sail from Deptford in the Hecla and the Fury in April 1821.
Parry sailed through the Hudson Straight and into Foxe Basin. (See Map.) Here he investigated Repulse Bay finding its name as apt as it had been when it had been discovered by Middleton 79 years earlier. Parry pushed north along the coast of the Melville peninsula, investigating bays and inlets to see if any might yield a passage west but to no avail. In October the ice closed in and the expedition was frozen in at ‘Winter Island’ for nine months. ...
When the ice allowed the ships free Parry made his way to what is now known as the Fury and Hecla Strait. Sadly for Parry the strait was choked with ice, although expeditions on foot did reveal a body of water to the west. After a second winter in the Arctic and no further luck in navigating the strait, which was still blocked by a solid ice barrier, Parry was forced to return to England due to lack of provisions."

"When the ice partially relented nine months later Parry searched for openings on the west side of Prince Regent Inlet but disaster struck when the Fury was forced aground by ice. Unable to repair the vessel it was abandoned (at what is now known as Fury Beach on Somerset Island) and sank. The Hecla, with two crews now on board, was obliged to return to England due to a strain on resources, even though Parry thought he could see clear water to the south. The voyage was not a complete failure. Although the search for the North-West Passage was hardly advanced the expedition collected much data on the position of the magnetic pole and arctic wildlife. Also, when John and James Clarke Ross reached Fury Beach a few years later the Fury’s abandoned stores would prove invaluable."

"In 1827 he attempted one of the earliest expeditions to the North Pole. He reached 82°45' North latitude, setting the record for human exploration farthest North that stood for nearly five decades before being surpassed at 83°20'26? by Albert Hastings Markham in 1875–1876."4

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¹ Wikipedia

² www.rmg.co.uk

³ www.rmg.co.uk

⁴ Wikipedia