As mappers we can link OSM objects to images, but the images themselves must be hosted elsewhere because OSM is not an image hosting service. You can upload the images of your parkour landmarks to wikimedia commons, flickr, or really any image hosting service, then update the tags for the POIs in OSM.
As for saving the POIs, most apps that use OSM have favorites or bookmarks, for example, I know OsmAnd does, and I’d bet Organic Maps does, too. I know OsmAnd also has a way of displaying the images linked to a POI, so what you’d do is bookmark your parkour landmarks in an app like OsmAnd, then you should be good to go.