Is it possible to run multiple thought processes concurrently? For example, let's say I have a friend sprinting a 50m distance and that friend wants me to time the race, but I don't have a stopwatch so I have to count "manually". Can I count while simultaneously thinking about how she'll do and while wondering what I'm going to have for dinner?
Parallel processes are often studied in a so called 'dual task' paradigm, where participants are drawing a picture and reciting a poem or, as in your example, counting and thinking about other things. 
Often this method is used to demonstrate limits in attention and find insights into how the brain works (in a serial versus parallel manner). Training is an important factor in this question. It has been shown that with a sufficient amount of training a lot of tasks can be done parallel very well - so if you sit there long enough and automize the counting the rest will follow.