In this context, you need to mentally separate the words. Your edit is accurate but the original is typical in spoken American English. "(Because I live closer,) I only have to walk down the street now / to (get to) my local supermarket..." It actually sounds better to place the second clause first and make two sentences using your edit: "Where my parents live you have to travel several hours to get to the nearest shops. Now I only have to walk down the street to my local supermarket, which is really convenient."

Now I only have to walk down the street to my local supermarket.. does that make more sense? It's a confusing placement of the word now in the original.