Do multi-channel opto-triacs with common pins exist?
As far as I know, optoisolators with triac outputs are not usually manufactured in the configuration you speak of.

If board space is a concern, you can select the 4-pin triac-output optoisolators instead of the 6-pin versions which would probably allow you to pack more optoisolators on the same board space.

Toshiba does manufacture dual and quad triac-output optoisolators, TLP525G-2 and TLP525G-4 respectively. The TLP525G-2 is a DIP-8, and the TLP525G-4 is a DIP-16. However, the individual channels within the package do not share any pins. Despite this, you might still be able to save board space by using a TLP525G-2 instead of two DIP-4 or DIP-6 optoisolators.

The dual channel version is available on Mouser and Digikey, I couldn't find a distributor that has the quad version in stock.