Hi James, an interesting post i too watched coast. I lived in Denmark for 12 years , greenkeeping, consulting and latterly designing and building courses . I worked for the DGU for over 2 years as their consultant so i have a good knowledge of most of the courses new and old over there. Around Copenhagen the oldest courses of note are Royal Copenhagen which is indeed in the deer park , they have just undergone a greens and bunkers rebuild by , Mackenzie and Ebert, this is the oldest club in Scandinavia and well worth a visit, it is the longest walk from car park to clubhouse you will ever find but worth the walk,
Rungsted is nice old parkland number four at Rungsted is in my opinion one of the best par fours in Golf, and there are some other great holes too.
Falster Golf Club is a great parkland course and hour and 20 south of Copenhagen great routing in a great setting. on Sjealland some of the other old courses worth a round are Helsingor, Gillileje, ,Koge and there are many more. New courses that play hard and fast are Vallo, Greve, Smorum(not great design but close to links conditions) you also have the new course at Skjondelsholm which is RTJ 2 its tuff to play though, and then the newest as Adam suggested is the Scandianvian club, access to this course is very restricted. Simons was a Hawtree course of note but i know they are having agronomic problems right now.
If you are going to Jutland over Fyn then visit Odense golf club on the way there are some other good courses on Fyn too St Knuds is a nice old course adjacent to the sea and the motorway.
On Jutland you have lots of options Aalborg, is a regular for European competitions at all levels a heathland course with some nice holes, you have Holsterbro, Veijle , Esbjerg, Fano, Hvide Klit, to name some of the old ones, but others like Volstrup, Lubker, Horsens, Kolding are all worth a visit.
If you let me know where you are going James i can try and help guide you,
Cheers
Chris