Steam or boil the potatoes in their skins (jackets) in salted water until just tender.
Drain the potatoes and let them cool off a bit. When they're just cool enough to handle, remove their skins. The skin should be loose enough to remove by simply rubbing with your hands, but if not you can scrape off the skin with a paring knife.
While they're still warm, cut the potatoes into slices or cubes and add them to a large mixing bowl. Pour over a generous glug of olive oil, season with salt and pepper to taste and, if you like, a squeeze of fresh lemon—just a few drops will do. Add the chopped onion and parsley, along with the tuna, broken into smallish bits and one or more of the optional ingredients if using. Mix everything together very gingerly, taking care not to break up the potato pieces.
Taste and adjust for seasoning. Let the salad rest for some time before serving at room temperature. If you like, save some of the tuna and parsley for topping, along with one or more of the suggested garnishes for a fancy touch.