Place the sundried tomatoes, basil, and garlic in a food processor and pulse to combine. Add the salt and pepper and 1/3 cup olive oil and puree until it forms a paste.
Add the Parmesan and the rest of the olive oil, then blend until smooth, adding additional oil if you feel the pesto is too thick. Taste and adjust the seasonings. Store for up to a week in the fridge.
Notes
Use sun-dried tomatoes packed in oil, which are generally much more pliable than the ones that are just sold dried, loose, or in bags. If you only can get dried tomatoes, you can soften them by putting them in a bowl and pouring over very hot water. Let sit for at least an hour, optimally two, at room temperature. Drain and use in the recipe. You may want to add another tablespoon or two of oil to the pesto to make up for the oil that would have been added with oil-packed sundried tomatoes. The oil used to store the sundried tomatoes may not be of very high quality, which is why I like to drain the sundried tomatoes and then add my own fresh olive oil. But taste the oil in the jar — if it seems fresh and has good flavor, then you can use that, and just add additional olive oil to make up the half cup called for.