Parker joined Costain from Mace in 2018 and first held the position of director, rail and airports before being promoted last May to become MD of transportation.
He moves after Costain issued a profit warning at the start of the year over a design dispute with the Welsh Government over its A465 Heads of the Valleys road project.
The firm said that 2019 expected profit would be cut in half to around £19m.
Costain said the rest of the business was performing in line with expectations.
The final cost and timescale for Costain’s A465 project remain uncertain. Work was due to finish in September 2018, but full completion is now expected to run to Spring 2021.
Parker has a breadth of experience in the construction and infrastructure sector spanning 30 years.
While at Mace has was seconded to Highways England to act as project director for the Trans-Pennine Tunnel scheme, and before that was interim director of the Smart Motorways Programme.
Costain is expected to announce his successor shortly and will unveil 2019 results tomorrow.