Pat the venison dry, trim off the hard fat and silver skin, and cut into 1.5-inch cubes. Toss the cubes in a bowl with the flour, 1 tsp salt, and black pepper until evenly coated, then shake off any excess flour.
Dice the onion, slice the carrots and celery 1/2 inch thick, and mince the garlic. Lightly crush the juniper berries, chop the parsley and refrigerate it, and leave the potatoes whole for now.
Heat 1 tbsp olive oil in the Dutch oven over medium-high heat. Sear half the venison in a single layer for 2 to 3 minutes per side until deeply browned, then move to a plate. Wipe out burnt flour, add another 1 tbsp oil, and repeat with the remaining venison.
Lower the heat to medium and add the final 1 tbsp olive oil. Cook the onion, carrots, and celery for 6 to 8 minutes until the onion turns translucent. Stir in the garlic for 30 seconds, then add the tomato paste and cook for 2 minutes until it darkens to a brick red.
Pour in the red wine and scrape the bottom of the pot with a wooden spoon to lift every browned bit. Simmer for 3 to 4 minutes until the wine has reduced by about half and the sharp alcohol smell has faded.
Add the beef broth, Worcestershire sauce, smoked paprika, thyme, bay leaves, and crushed juniper berries. Return the seared venison and any juices to the pot, bring to a gentle boil, then reduce the heat to low, cover, and simmer for 2 hours, stirring every 30 minutes. Test a cube with a fork at the 2-hour mark and simmer another 20 to 30 minutes if still firm.
About 15 minutes before the meat is ready, scrub and cut the potatoes into 1-inch chunks. Stir them into the stew once the venison is nearly fork-tender, cover, and simmer for 30 minutes until the potatoes are fork-tender. Remove the lid for the last 10 minutes if the gravy looks thin.
Remove and discard the bay leaves. Taste and add salt until the flavors are balanced, usually about 1/2 tsp more. Turn off the heat and let the stew rest for 10 minutes, then ladle into warm bowls and sprinkle with the chopped parsley.