Is there a way to tell which spaceship parts the other civs have built?
Yes! There is a way to see this.
Just open the victory progress screen and you'll see something like the picture below 
If you then click on details for the science victory you'll see this 
This will show you how far every nation has progressed towards completing the spaceship.
On my screenshots all nations are still unknown because I made this screenshots in a new game on turn 1.