You'll want to enter the author name manually, then check "Suppress author" in the citation entry dialogue in OpenOffice (or Word) when you insert the date part.

It just seems to defy logic that in order to get valid APA format I have to override Zotero's standard referencing and put author names in by hand. Isn't doing this correctly the reason to use Zotero? Others seem not to have this problem.

Surely there's another solution to this that doesn't involve me re-inserting every reference in my research paper and typing in the author names!

Thanks
Graham

just to highlight that (Author, Date) is valid APA, look here:
http://owl.english.purdue.edu/owl/resource/560/03/
it's of course easy to adjust the style for your needs by changing
<sort>
<key macro="author"/>
<key macro="issued-year"/>
</sort>
<layout prefix="(" suffix=")" delimiter="; ">
<group delimiter=", ">
<text macro="author-short"/>
<text macro="issued-year"/>
<text macro="citation-locator"/>
</group>
</layout>
</citation>

to

<sort>
<key macro="author"/>
<key macro="issued-year"/>
</sort>
<layout>
<text macro="author-short"/>
<group delimiter=", " prefix="(" suffix=")">
<text macro="issued-year"/>
<text macro="citation-locator"/>
</group>
</layout>
</citation>

http://www.zotero.org/support/csl_simple_edits
You also have to change the and="symbol" in the author-short macro to and="text".

I personally think that's a terrible idea for a whole bunch of reasons (among them what ajlyon says, the fact that this will make it impossible to convert to a footnoted style like Chicago, the fact that APA requires and ampersand in citations, but a text "and" in the text etc.), but if that's what you want it's a two minute change.

I would like to take up the discussion on this again, as I now face similar problems. If it has been resolved, I would be very thankful for a short note as to how.
The problem is, that (Author, Year) is NOT a valid in-text citation considering APA Style, 6th edtition. At least, if a direct reference to the work of the author(s) is included in the text, it is not.
You may kindly verify that by checking the publicly available paper on http://www.tandfonline.com/doi/full/10.1080/02699930903132496.
Compare, e.g., "Analogous distinctions highlighting the source of the affective feeling have recently been proposed in other fields (Laney, Heuer, & Reisberg, 2003; Pham, 2007)." where no direct reference to the paper of the authors is made and thus (Author, Year) is valid to "Seminal research by Butler and Mathews (1983, 1987) used self-report methodology, and demonstrated clear differences in interpretation between anxious and non-anxious individuals." where direct reference to the work of the authors is made and thus it is mandatory to cite in the Author (Year) style.
My suggestion (or rather question if there is already existing such a CSL) is to split up APA citation in zotero to in-text direct reference and in-text indirect reference.
I am aware that this does not solve the genitive problem noted above, but adding a "'s" at some points is infinitely easier than changing nearly every citation from (Author, Year) to Author (Year).
Thank you for your patience in reading all of this, I hope someone can point to a solution of this, as time for my thesis is running out and I would be very happy if I did not have to change the style myself completely.