The Metro board approved a plan yesterday allowing Montgomery County to build a two-story parking garage at the Shady Grove Red Line station, adding 800 spaces to a 3,500-car lot that on weekdays fills up by 9 a.m.

Construction is scheduled to begin in January and be finished by December 1988, to coincide with the planned completion of I-370, a four-lane highway linking I-270 with the Shady Grove station.

The I-370 link is expected to relieve congestion on I-270, assuming that there is adequate parking at Shady Grove. County officials had estimated that at least 1,000 additional spaces were needed by 1990.

County Executive Sidney Kramer proposed last December that the county add 1,650 spaces at the station by building two parking garages. The new garage represents the first phase of that project, and a second structure will be built if needed.

"We hope this gets more people off the roads and onto the transit system," said Carlton R. Sickles, one of Montgomery County's representatives on the Metro board.

The county offered to pay for the design, construction and operation of the structure, to be built on the west part of Lot 2. The cost of designing and building the first garage was estimated at $5.5 million, said Edward A. Daniel, an assistant to the county's director of transportation.

The County Council has approved money for the design and will soon be asked to appropriate money for construction, Daniel said, adding that Maryland officials have agreed to contribute $3 million.

Metro's consent was needed because the structure will be built on Metro property.

About 550 parking spaces at Shady Grove will be lost for the year-long construction period, during which the county plans to allow parking along an unopened section of I-370 and to provide free shuttle buses to the station.

The board action comes as the transit agency and several local jurisdictions are addressing the problem of insufficient parking at many suburban Metro stations.

Last December, Metro, Amtrak and Prince George's County opened a $9 million parking garage at the New Carrollton station. The county, Metro and the Federal Railway Administration shared the cost.

Fairfax and Arlington counties asked Metro to raise parking fees at five stations and lower bus fares on some routes to encourage commuters to leave their cars at home.

The pricing plan was begun in March as a year's experiment. However, Metro Chairman Joseph Alexander, a Fairfax supervisor, has often said that garages are the only long-term solution. The problem, however, is finding the money to build them when other transportation projects are competing for funds, he has said.

"There's no Metro money available" for parking structures, Sickles said. "I think other jurisdictions will have to follow" Montgomery County's example by paying for their own parking structures, he said.

The Metro board is expected to consider in a few weeks a proposal by Montgomery County to add 125 parking spaces at the Rockville station, where the 550 spaces fill by 7:30 a.m. on weekdays, Daniel said.

In other action, the board:Voted to spend $576,024 for trucks, tractors and other equipment to remove snow from Metro driveways and parking lots, which were snowed in last winter.

Voted to seek an investigator to examine some minority-owned businesses competing for Metro contracts to ensure that the companies are owned by minority group members. An investigator hired in 1986 disqualified five of eight firms examined, a Metro official said.