In attempting to solve the problems caused by a lowering of the price of oil, oil companies operating in the North Sea have taken a variety of approaches, which includes their reducing employment, using new technology to pump oil more efficiently from smaller fields, and finding innovative ways to cut the cost of building and operating platforms.

Understanding the sentence first is the most important thing here the oils companies have taken variety of approaches- hence the approaches include
Now review the questions

A. which includes their reducing employment, using new technology to pump oil
more efficiently
WRONG since approaches is plural word

B. which includes reducing employment, using new technology to be more efficient in pumping oil
WRONG same reason

C. which include reducing employment, using new technology to pump oil more efficiently
CORRECT no parallelism error, no verb error

D. which include the reduction of employment, their using new technology to pump oil more efficient
WRONG parallelism error- reduction of employment is not parallel to using and finding

E. including a reduction of employment,
WRONG same parallelism error