While many recipes call for using either parchment paper or a silicone baking mat like a Silpat, the two aren't 100-percent interchangeable. Both use food-grade silicone to keep food from sticking to a surface, but they differ in some crucial areas and should be used accordingly.
A silicone baking mat is much thicker than a sheet of parchment paper and thus insulates the pan it's laid atop. This can be a boon when baking delicate cookies, like these French macarons. On the flip side, use parchment paper if browning on the underside is desired, like when roasting vegetables or baking sugar cookies.
Because it is disposable, parchment can be used in many recipes where a silicone baking mat cannot: It can be cut to size to line cake pans and casserole dishes. Additionally, parchment paper can be cut to size and folded into an airtight pouch around delicate items like seafood to cook en papillote.
In most other applications, either nonstick liner can be used; the choice is up to you. Some cooks prefer to use a silicone baking mat, as it is reusable and thus eco-friendly. Others prefer parchment paper for its ease of use (simply recycle a sheet after use instead of scrubbing it clean).