  Observations were made of the open clusters M35 and NGC 2158 during the
initial K2 campaign (C0). Reducing these data to high-precision photometric
time-series is challenging due to the wide point spread function (PSF) and the
blending of stellar light in such dense regions. We developed an
image-subtraction-based K2 reduction pipeline that is applicable to both
crowded and sparse stellar fields. We applied our pipeline to the data-rich C0
K2 super-stamp, containing the two open clusters, as well as to the neighboring
postage stamps. In this paper, we present our image subtraction reduction
pipeline and demonstrate that this technique achieves ultra-high photometric
precision for sources in the C0 super-stamp. We extract the raw light curves of
3960 stars taken from the UCAC4 and EPIC catalogs and de-trend them for
systematic effects. We compare our photometric results with the prior
reductions published in the literature. For detrended, TFA-corrected sources in
the 12--12.25 $\rm K_{p}$ magnitude range, we achieve a best 6.5 hour window
running rms of 35 ppm falling to 100 ppm for fainter stars in the 14--14.25 $
\rm K_{p}$ magnitude range. For stars with $\rm K_{p}> 14$, our detrended and
6.5 hour binned light curves achieve the highest photometric precision.
Moreover, all our TFA-corrected sources have higher precision on all time
scales investigated. This work represents the first published image subtraction
analysis of a K2 super-stamp. This method will be particularly useful for
analyzing the Galactic bulge observations carried out during K2 campaign 9. The
raw light curves and the final results of our detrending processes are publicly
available at \url{http://k2.hatsurveys.org/archive/}.
