Hi,
What time of the day will your parents be in Copenhagen? I know the city very well. I was born in Malmö, right across the water, and grew up there. I have been to Copenhagen hundreds of times. For a couple of years (long time ago) I worked at Copenhagen Kastrup Airport.
If your parents only have 4-5 hours, my advice would be for them to just walk around. When they exit the train station, the side entrance to Tivoli amusement park is RIGHT ACROSS the street. They could easily spend the entire time in Tivoli but I would suggest they limit their time their to 2-2.5 hours. If they exit Tivoli on the opposite side from where they entered, they will have to cross a major street (H.C. Andersens Boulevard) and they will then be in a large open square: Rådhuspladsen (=City Hall Square). City Hall, an imposing building, will be on their right (impossible to miss). Cross the square, and the famous "Strøget" pedestrian street is right there. "Strøget" was the very first pedestrian street anywhere (as far as I know) when it opened to pedestrians only back in the early 1960s. It's actually a series of connecting, narrow, streets that run from Rådhuspladsen to Kongens Nytorv, about 2 miles/3 km away. The official name of the street when you enter from Rådhuspladsen is Frederiksberggade. It's a very pleasant and easy walk, and at a leisurely place, and stopping at a few places (shops) on the way, you can easily walk the entire length of Strøget in an hour or an hour and a half. If your parents spend a couple of hours and then walk Strøget they will have utilized their time in Copenhagen very well. When they get to Kongens Nytorv, simply take a taxi back to the train station. Forget about a bus card. They won't have the time to take advantage of that and once they get to Kongens Nytorv they don't want to try to figure out what bus to take etc etc. A taxi will be a bit more expensive than the bus obviously, but without the hassles. And MUCH quicker.
If you, or they, have any further questions, let me know!