You have a lot of options for spending the night on a bike tour in Denmark. On the maps here at ruter.dk you find hotels, inns, hostels, bed & breakfast, camping, holiday resorts, shelters and free camping areas.
In the summer months, it may be a good idea to book accommodation in advance, unless you are cycling with a tent. Most campsites will always find room for one more tent. Please note that many shelters can be booked via udinaturen.dk.
At many accommodations you can park the bike safely for the night. Ask, if you’re in doubt of where to park. Experience shows that a good solution almost always can be found.
If you ride an electric bike and need to charge the battery overnight, you better ask permission first. There is a small risk of fire when charging batteries, and the accommodation may have a special place for it. If you spend the night in a shelter, you must plan a longer coffee stop to recharge your battery.
Many Danes offer free or very cheap private accommodation. Check brugminbaghave.dk and warmshowers.org.