Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and reserve about 1 cup of the pasta water.
In a 14-inch skillet, heat 1/4 cup oil over medium-high heat. Add the onion and cook, stirring frequently, until soft, about 5 minutes. Add the garlic and cook for 30 seconds until aromatic. Add the tuna to the skillet and, using a fork, break into chunks. Add the cherry tomatoes, artichoke hearts, capers, and thyme. Cook, stirring occasionally, until the tomatoes begin to soften, about 8 to 10 minutes. Add the pasta, the remaining 1/4 cup olive oil, and the parsley. Toss until all the ingredients are coated, adding a little pasta water, if needed, to thin out the sauce. Season with salt and pepper, to taste. Transfer the pasta to a serving bowl and serve warm or at room temperature.