Clipping an image to the extents of a bounding box is very easy to do with the gdal_translate utility from GDAL. The documentation for it is here. You'd want to use the "-projwin" option like this,
gdal_translate -projwin ulx uly lrx lry infile outfile
For the URL you gave, I would assume the following should work (Note that your ESRI command specifies LL and UR, while GDAL uses UL and LR)
$ gdal_translate -projwin -9562211.80152863 4626760.02856153 -9519931.21541649 4584479.44244939 <source_image> <output_image>
This would give you an output_image that is clipped to the extents of the bounding box. You should note that you cannot control the size of the resulting image when doing this. You get back as many pixels are needed to fill the extent. If you want to make the output_image a specific size, you can do that as a second step with gdalwarp and specify whatever resampling type you want to use.
GDAL has an excellent Python API, so you could combine both operations and glue the output into your processing setup, but I have to wonder whether this is really what you want? If all you need to do is display your image over a web map, it would be far simpler to setup Mapserver, point it at your image and either use WMS, or a tile request to bring the data into your mapping API.
Roger