The correct saying is :

J’ai beaucoup parlé hier.

"Parler" is already the general action of saying words. So you can’t really quantify the result.

You could however put "beaucoup" after the verb "dire" :

J’ai dit beaucoup de bêtises.

As "dire" doesn’t necessarily imply words (it could be prayers, speeches, etc.).

In most cases (if not all), "beaucoup" precedes the verb if the object isn’t specified :

J’ai beaucoup mangé.

J’ai mangé beaucoup de bacon.