I am supporting the idea of anything mounted - horsed for WW1 and WW2 and even beyond (Rhodesia 1970's), but I have my doubts that there were partisans/guerrillas mounted on horses in WW2.
Most partisans/guerrillas remained in remote areas where traveling is restricted by natural barriers.
Significant partisan/guerrilla formations were present in Balkans where they used mountains and forests for their hideout.
In Russia/Ukraine they were mainly present in marches...
Mountains, marches, forests - none of those are place where one can use horses in significant number... Furthermore, partisans/guerrillas were starving most of the time and horses would be huge expense and hard to hide. Horses there would usually end up as additional nutrition...
Occupation forces used large amount of horsed units - but that was done in order to patrol communications and prevent sabotage. So, I would vote for more Axis cavalry! It is true that most of it ended up in actions that were not the ones reserved for cavalry. (22nd Maria Theresia and 8th Florian Geyer SS divisions were destroyed in the siege of Budapest, while 37th SS cavalry division Luetzow was formed from survivors. 7th Prinz Eugen SS mountain division disbanded her cavalry squadron later in the war. As a rule, these units were formed by ethnic Germans recruited in Serbia, Hungary and Romania.)
As of partisans/guerrillas I'd rather see Polish, French and Italian - but to my knowledge they had no cavalry what so ever...