Patrick Star's second rock is a rock Patrick uses to disguise SpongeBob. It is only seen in the episode "Suds."
The outside of Patrick's second house appears to be a replica of Patrick's regular house. It even appears to be the same size, but instead covered in dust.
After Patrick hears that Sandy has arrived, Patrick uses a second rock to disguise SpongeBob with, so that Sandy would not take him to the doctor. Sandy asks "Since when did you have two houses," to which Patrick replies "Since I ran out of space to put my stuff." Sandy then asks "Since when does your house have feet?" to which Patrick replies "This is my mobile home." SpongeBob then sneezes, causing the rock to explode.