Is the apostrophe in the right place in the following sentence?
Pendleton, et al. (2002)’s research implies that extension of treatment allows for greater weight loss.
Logic suggests that, but it's pretty ugly. No one is coercing you into using the ’s mark of the possessive, and it would be much nicer to rewrite it as:
The research of Pendleton et al. (2002) implies that…
If this is academic writing, then there is no one answer that will work in any situation.
The answer actually depends on the specific field you are writing in. In my field (linguistics), there is a standard way that this is done, and it is like the following, e.g.:
Chomsky's (1957) analysis of...
Dupoux et al.'s (1999) finding of...
Other fields might specifically demand that you do it the other way, but in linguistics, this is just what you do. If you do it the other way in a peer-reviewed publication, it will be corrected. It's not really a matter of logic, but of formality and tradition.
(Also: the best way to know what happens in your field? Outside of asking someone knowledgable in the field, you can look at two or three papers and probably find at least a few examples.)