Heat the olive oil in a large, heavy-bottomed stockpot over medium-low heat. Once hot, add the leeks, garlic, and a pinch of salt, and sweat until they begin to soften, 7 to 8 minutes. Add the carrots, potatoes, and green beans, and continue to cook for 4 to 5 more minutes, stirring occasionally.
Add the broth, increase the heat to high, and bring to a simmer. Once simmering, add the tomatoes, corn kernels, and pepper. Reduce heat to low, cover, and cook until the vegetables are fork-tender, 25 to 30 minutes. Remove from heat and add the parsley and lemon juice. Season to taste with kosher salt. Serve immediately.