Dench will join the board as chair on 14 March, after Dennis Holt, who has been chair since October 2014, steps down to retire in May.
The incoming chair held the same role at Paragon since 2007. He will hold both roles until Paragon finds a successor.
Read more: Dennis Holt hired as Co-op Bank chairman
Holt's move to step down was announced after Co-op Bank last year secured a rescue package deal with its US hedge fund investors in June. The investors, including Blue Mountain Capital Management, Cyrus Capital Partners, GoldenTree Asset Management and Silver Point, wrote off £443m in debt as well as running a £250m fund-raising.
The Co-op Bank has "been through challenging times in the past four years as it has worked hard to fix the legacy issues that led to the crisis it faced in late 2013," Holt said, referring to the dramatic removal of its former chair, Paul Flowers. "It now has solid foundations and a clear pathway to sustainable profitability and robust capital resilience."
Dench, who joined Paragon's board in 2004, chaired it through the financial crisis and oversaw its application for a banking licence. He previously served on the boards of Axa UK and Ireland from 2004, chairing Axa Ireland from 2007 to 2017. Prior to that, he spent 28 years at Barclays, in a number of senior executive roles across the group in the UK, US and Australia.
Liam Coleman, chief executive, said the bank still has "much to do" after last year's recapitalisation, welcoming Dench's arrival.