What to use instead of cling film to wrap pastry
It is less useful than what you think

Frame challenge incoming...

Cling film is very light and made especially for such purposes. The environmental damage is extremely low - which limits what alternatives you can choose. Most alternatives (including those already mentioned in the other answers) will be so much more resource demanding to make, dispose or recycle for it to be worth it. Baking paper, while extremely useful - is full of silicones, so don't put it in the "paper" bin... and it is more resource demanding to make. Teflon, cloth, etc - same deal!

Aluminium foil, for instance. 75% of all Aluminium refined and made is still in use. Because of the extremely friendly "recyclability" of aluminium, it is a very good material. Just not for aluminium foil, it is rarely recycled. People just throw it in the trash. We destroy a valuable resource. 

Cling film, on the other hand is  dirt cheap to make and can safely be incinerated. If you worry about polluting the sea - I'd see if there are any alternatives to landfilling your waste in your community. As for carbon emissions - Walk once to the store instead of driving, and you'll be in the green even if you consider a life time supply of cling film. 

Note: Some cling films are made with PVC, I'd consider switching to the less clingy, but better overall alternative LDPE. PVC contains chlorides and while the environmental impact is still low, if you go for LDPE or similar it goes to negligible. Check the box it came in to find which you have