If you are searching for a pre-scheduled run, lets say on Saturdays I must disappoint you. Currently there are no “Park-Runs” in Ljubljana. There are some local clubs though which organize prescheduled group running, but more in a sense of training or preparation for Marathons, where you pay a fee to take part for a month, season, etc.
Central Ljubljana has one park appropriate for running, which is accessible from all central Hotels, here is where most of the locals go for a run, walk, hike,… Tivoli Park is located on the western part of City Center, stretching to the Šiška District to the north, the Vič District to the south, and the Rožnik District to the west. Several notable buildings and art works stand in the park. Since 1984, the park has been protected as part of Tivoli–Rožnik Hill–Šiška Hill Nature Park. It is home to a variety of bird species. (More about Tivoli Park)
If you are staying in City Center I would suggest you start to run next to Modern Art Gallery take the underpass and enter Tivoli there you can decide to follow one of the paths that take you to the Cankarjev Vrh. Map below shows the most popular path, there also many other paths and destinations one could take...
On our sightrunning routs we also visit Tivoli Park and run pass quite a few Landmarks there too. Routes that take us there:
Google navigation link for the map below.