Which visa types enable a foreigner to work in Thailand?

If they have a work permit and a "work visa" from one company, are they allowed to do part-time work at another company?

While there is no such thing, as a 'work visa' most non-immigrant visa categories allow to apply for a WP.

Most common sub-categories would be:

B, O, D, M, F and R but also M, EX and in some cases Ed.

For details have a look here:

http://www.thaivisa.com/visa/non-immigrant-visa.html

A work permit allows work in Thailand - not a visa.

A work permit can be issued if you have a non immigrant visa but normally those who have for retirement will not be issued a work permit.

All work/locations must be on work permit.