connect computer to magnetic exercise bike
Physical Connector

Parts list in the manual shows part 18 - middle wire.  If you had spare 1x or 2x of those, you would have enough material to build the bifurcated cable to connect to a computer.  You may be able to buy it from York Fitness, pretending that you need it for repairs.  (In fact, there is a parts order form in the end of the manual.)

Raw Sensor Signals

Parts list in the manual shows parts 17 and 28 and mentions sensors.  It's possible that the console takes raw sensor inputs.  These may be analog signals.  

In this case, you would need a data acquisition (DAQ) device for the PC.  There are lots of small USB-based DAQ boxes on the market.  To name a few: National Instruments USB-6008, LabJack, DATAQ.  Some of them have API for .NET.

Obviously, you may have to do some reverse-engineering.  Get to the sensors, try to identify what they are, look-up datasheets for them.  Observe the signals at the connector pins with an oscilloscope.

CSAFE

A more remote possibility is that your exercise bike supports CSAFE (Communications Specification for Fitness Equipment).  It's an RS232-based protocol.  The specification is public.