Grate the carrot and parsnip into thin shreds, then finely dice the onion and mince the garlic cloves. Heat a large skillet over medium heat with a touch of vegetable oil. Add the grated carrot, parsnip, and diced onion, cooking for about 5-7 minutes until the vegetables soften and release their moisture. Add the minced garlic and continue cooking for another 2-3 minutes until the mixture becomes dry (this concentrates the flavors). Transfer the cooked vegetables to a bowl and let them cool slightly before proceeding—this prevents the eggs from cooking when mixed in.
To the cooled vegetable mixture from Step 1, add the ground beef, ground pork, fresh parsley, eggs, crushed crackers, potato starch, salt, and pepper. Mix gently with your hands until just combined—overmixing will make the meatballs dense and tough. Divide the mixture into roughly 1½-inch balls and place them on a plate. I find it helpful to wet my hands slightly with water while shaping; it prevents sticking and gives you cleaner meatballs.
Heat 2-3 tablespoons of vegetable oil in a large skillet over medium-high heat. Working in batches if needed to avoid overcrowding, add the shaped meatballs and brown them on all sides for about 8-10 minutes total, turning occasionally for even color. You're not cooking them through at this stage—just developing a golden crust for flavor. Transfer the browned meatballs to a clean plate.
In the same skillet (don't clean it—the browned bits add flavor), whisk together the beef broth and the potato starch that has been mixed with water. Bring this mixture to a gentle simmer over medium heat, stirring occasionally for about 5-7 minutes until it thickens slightly and becomes glossy. Stir in the fresh parsley and season with salt and pepper to taste. The gravy should coat the back of a spoon lightly—this creates that classic Irish stew consistency.
Return the browned meatballs from Step 3 to the skillet with the gravy from Step 4. Reduce heat to medium-low and gently simmer for 10-15 minutes, stirring occasionally to coat the meatballs evenly. I like to let them simmer uncovered so the gravy reduces slightly and the flavors meld together beautifully. The meatballs are ready when they're cooked through and the gravy has the silky texture of a proper Irish stew. Serve warm directly from the skillet, or transfer to a slow cooker on the warm setting if making ahead.