Given you only need to use versions of Excel 2007 and later, you could use SUMIFS()
e.g.
In your sample
=SUMIFS(J2:J31,A2:A31,A2,I2:I31,">"&I2,I2:I31,"<"&I31)
This will give you the total parts sold between, but not including, 15-Jan-10 and 05-Feb-10
.
i.e.
29-Jan-10

Hope this helps