I think starting from FF8, Squaresoft doesn't put their movies into separate .STR files anymore. They usually jam it up into one file, prolly to save space and to prevent us from ripping the MDECs. To view them, you can use PSMPlay, but you first have to use the "Analyze CD" option.
It's under "File->Analyze CDRom"... I've tried this before with FF8 and it worked like a charm.
If that doesn't work, get PSXPlay instead.... but this one's not free. You gotta register first before it would play the entire movie.