Which year do I use for an in-text APA citation? The first year is when the data was collected: 2013
The second year is when the paper was written: 2015
The third year is when the paper was published: 2017


"In total, 121 women and 61 men completed the experiment over the course of several months across 2013." is where I am getting the 2013 from, so if I was to say, in a study conducted in 2013, blah blah blah, very specific to the data itself, and I do an in-text citation in APA format, which year do I use?
It needs to be the year of publication in the in-text citation (same for the reference at the end of the paper). This is backed up by the Purdue OWL:


  When using APA format, follow the author-date method of in-text citation. This means that the author's last name and the year of publication for the source should appear in the text, for example, (Jones, 1998), and a complete reference should appear in the reference list at the end of the paper.


https://owl.purdue.edu/owl/research_and_citation/apa_style/apa_formatting_and_style_guide/in_text_citations_the_basics.html

If the date the survey was conducted is important then you should definitely mention it in the surrounding text like you’re doing. 

Due to the fact that writing a paper then going through peer review takes time, it’s not unusual to see some delay between the date the experiment was done and the date the paper was published. Even though this does seem like a bigger gap than usual, I probably wouldn’t even question it if I was reading it.