I think that the DVD drive may not be reading the DVD reliably.
Make sure that the surface of the DVD is clean, and no visible scratches.
Use a "can-o-air" on the DVD drive.
You can also try a cleaning DVD (the kind with the little brushes imbedded in the disk)
Or, your MacBook Pro is at least 8 years old, and an original DVD drive may just not read the disk very well, and needs replacing. With that in mind, you could then try an external DVD drive for the install.
Another method (which is what I would do) is to get an 8GB USB flash drive. Erase it, and make sure it is formatted as a GUID volume.
While in Disk Utility, click the USB drive, and choose the Restore tab.
Put your DVD in the drive, then drag the drive to the Source line in that Restore window.
Drag your USB flash drive to the Destination line.
Click Restore. That will take anywhere from 10 to 30 minutes to complete.
The flash drive then will be a bootable Snow Leopard installer, allowing you to install without worrying about whether the drive will read all the installer files without error.
If the Restore won't complete without an error, then make the installer USB drive on a different Mac that will read the DVD completely.