Neither side managed to turn first-half possession into scoring chances and Posh, with a greater share of the ball, struggled to release twin strikers Craig Mackail-Smith and McLean.
Only defender Jamie Day seriously tested Phil Barnes with the Grimsby keeper doing well to reach a fiercely-struck 25-yard free-kick.
Grimsby, with just Isaiah Rankin committed up front, found progress equally difficult although Jamie Clarke wasted their one decent opening, lifting a free-kick high over the bar after Rankin had been fouled by Charlie Lee on the edge of the box.
After the break Grimsby looked a different side. Nick Fenton rose high to rattle the home bar with a header from Ciaran Toner's free-kick and, after Danny Boshell saw a shot blocked, Posh's Shwan Jalal did well to keep out Peter Till's drive diving to his left.
In more Grimsby pressure Kieran Charnock punched away a Toner centre and was shown a yellow card by referee Mike Dean before one-time Spurs youngster Toner gave Jalal no chance with a confidently-struck 59th-minute penalty.
Posh drew level five minutes later with second-half substitute George Boyd sending in a towering effort from way out on the left touchline with the ball sailing over everyone, Barnes included, and into the far corner of the net.
McLean edged Posh in front with an 80th-minute penalty after Till looked harshly treated when ruled to have brought down an already stumbling Day.