Just want to throw in my 2 cents.
As a person who lives not far from Lyngby, and frequently goes through the main road, i think it could be worth trying out the market out here.
I personally feel like there's not many options up here besides your typical Asian/Thai massages.
Traffic in central Lyngby can be pretty chaotic at the moment and for the forseeable future with the tramway tracks being built.
Parking options varies from where in Lyngby the apartment is in. If it's in the industrial area or DTU area the parking options will easier to find than on the main street.
Else if there's no free spaces on the side of the road, there's always the option to park in Lyngby Storcenter.