The Aberdeen-based oil and gas firm said it had been provisionally awarded the blocks in the latest UK licensing round, which won a strong response from the industry.
The blocks contain a gas discovery and a number of prospects. They are in the Southern North Sea, where Parkmead has already enjoyed success.
Mr Cross got to know the area well during his time at Dana Petroleum.
He made around £57m when Dana was acquired by Korea National Oil Corporation for £1.9 billion in 2010.
Dana Petroleum said last week that Marcus Richards had decided to step down after three years as chief executive.
Mr Cross said: "The new blocks provide an excellent addition to our Southern Gas Basin portfolio, where Parkmead already has the Platypus gas field and recently made an important gas discovery with the Pharos exploration well."
The Platypus find was made by Dana during Mr Cross' time in charge of the company. Parkmead acquired an interest in Platypus when it bought stakes in three blocks in the Southern North Sea from ExxonMobil in 2011. Mr Cross noted Dana had been awarded a total of 30 blocks in the UK's 27th licensing round.
The five new blocks are covered by two licences that were included in the second tranche of awards.
Dana will operate the new blocks, giving it more control over the pace of developments than it would gain through a non-operated interest. North Sea-focused Ithaca Energy and Bridge Energy are partners in four of the blocks. Bridge, recently acquired by Norway's Spike Exploration for £103m, also has an interest in the other block.
Parkmead and Dana Petroleum announced the Pharos discovery last month.