Mr Hunt announced his decision to quit as MP for Stoke-on-Trent Central last week to become the director of London's Victoria and Albert Museum.
Mr Reed is leaving the Copeland constituency to take up a post at the Sellafield nuclear plant.
The MPs are expected to leave the House of Commons early next week.
In his letter of resignation, Mr Reed said his last act as an MP was "to implore" Copeland voters to back Gillian Troughton, a former hospital doctor and Labour's candidate for the seat.
"This by-election is about the Tory government's dangerous proposals for the NHS," he wrote.
"The Tories have created a crisis in the NHS and, as a result, the future of West Cumberland Hospital and the services provided by Keswick Hospital are under threat."
Ms Troughton, who said it was "an honour" to be selected by Labour, faces a tough battle in the West Cumbrian seat where her party's majority over the Conservatives was cut to 2,564 at the last general election.
Labour leader Jeremy Corbyn said: "Gillian is a local councillor with a strong track record of getting things done for her community. She has campaigned tirelessly to maintain local hospital services.
"As a St John's blue light ambulance driver, Gillian has seen first-hand the extent of the crisis caused by this Conservative government, which has chosen to fund tax cuts for the wealthiest instead of our health service."
The Conservatives said the selection was a "victory for the hard left", describing Ms Troughton  - who voted to remain in the EU referendum and is a supporter of nuclear power - as "nothing more than a Corbyn puppet".
However, her choice has been seen elsewhere as a boost for moderates, as Ms Troughton was one of 1,000 Labour councillors who signed a letter of support for Owen Smith in his unsuccessful attempt to unseat Mr Corbyn as leader last year.
Mr Reed held Copeland with 16,750 votes (42%) in 2015, ahead of the Conservatives on 14,186 (36%), UKIP on 6,148 (15%) and the Liberal Democrats on 1,368 (3%).
UKIP has chosen NHS worker Fiona Mills to contest the seat. She stood in Carlisle at the 2015 election, coming third with 12% of the vote.
"Copeland voted 62% to leave the EU," she said. "What better way to send a message to the government that we want a full Brexit as soon as possible than to return a UKIP MP."
Rebecca Hanson has been chosen to fight the seat for the Liberal Democrats.
Labour is due to choose a candidate on 25 January to fight Stoke-on-Trent Central, in a by-election triggered by historian Mr Hunt's appointment as director of the V&A Museum.
Despite being a traditional Labour safe seat, Mr Corbyn faces an awkward contest in a city which voted overwhelmingly Leave in last year's EU referendum.
UKIP - which took second place in the 2015 general election - is due to announce its candidate on Saturday, amid widespread expectation that leader Paul Nuttall will fight the seat in the hope of becoming his party's second MP in the Commons.
By-elections sparked by the resignations of Labour MPs Tristram Hunt and Jamie Reed are to be held on 23 February, the party has said.