There have been several papers on the parity of the partition function in arithmetic progressions. I don't know whether any of them cite computational evidence one way or the other, but it might be worth having a look (or just writing to Ken Ono). Cutting and pasting from Math Reviews, 

MR1844553 (2002f:11139) 
Boylan, Matthew(1-WI); Ono, Ken(1-WI)
Parity of the partition function in arithmetic progressions. II. (English summary) 
Bull. London Math. Soc. 33 (2001), no. 5, 558--564. 


MR1945975 (2003j:11128) 
Subbarao, M. V.(3-AB)
Partitions---some parity problems and results. Proceedings of the Second International Conference of the Society for Special Functions and their Applications (SSFA) (Lucknow, 2001), 59--65, Soc. Spec. Funct. Appl., Chennai, 200?. 


MR1816213 (2002i:11102) 
Ahlgren, Scott(1-PAS)
Distribution of parity of the partition function in arithmetic progressions. (English summary) 
Indag. Math. (N.S.) 10 (1999), no. 2, 173--181. 


MR1384904 (97e:11131) 
Ono, Ken(1-IASP)
Parity of the partition function in arithmetic progressions. 
J. Reine Angew. Math. 472 (1996), 1--15. 

EDIT: I have found a paper which seems to present numerical results on $p(n)$ for $n$ outside of certain arithmetic progressions. Neil Calkin et al., Computing the integer partition function, Math Comp 76 (2007) 1619-1638, freely available on the web at http://www.ams.org/journals/mcom/2007-76-259/S0025-5718-07-01966-7/S0025-5718-07-01966-7.pdf