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.
I have to partly disagree with schultem here-- he is right that the dual task paradigm is used to study multitasking, but the idea that we can truly do two cognitive tasks at the same time is still contentious. In particular, the opposing view might say that we can't really do two cognitive processes at the same time, but we are in fact switching between the two tasks very rapidly. To make a an analogy, consider a computer with a single-core CPU: although behaviorally it seems like you are running multiple processes at the same time, all applications still depend on a single bottleneck.
A typical PRP task has the subject perform two reaction time tasks. For instance, he may have to hit a button with his right hand in response to a high pitched tone, and hit a button with his left hand in response to a low pitched tone. At the same time, he performs another concurrent task-- e.g., saying "blue" in response to a blue stimulus, and "green" in response to a green stimulus. By altering the stimulus onset asynchrony (SOA) (that is, the time from the start of the tone to the start of the visual stimuli), we can examine what operations people are able to perform in parallel.
Each task can be broken down into three components: perceptual analysis, response selection (cognitive), and response processing. A prominent theory emerging from the results of many PRP tasks is that we are able to truly do two operations in parallel if they require different modes: for instance, we can do perceptual analysis of a visual stimuli and auditory tone at the same time, and we can make a manual and vocal response at the same time, however we cannot do two response selections at the same time because they essentially require the same cognitive resources (single bottleneck).
This image from wikipedia describes what I'm talking about. The response selection from the second task must wait for the response selection from the first task to finish before it begins. The result is that, at short SOA, the total time taken to respond to both stimuli is the same-- as long as perceptual analysis of the second task ends before the response selection from the first task.
So in sum, yes, it might be possible to do the two tasks you are talking about in parallel, especially with practice. But on a more fundamental level, you may really just be doing very rapid multitasking. If you're interested in this theory, you may also want to read about threaded cognition, which demonstrates this effect for complex tasks.
I experience thinking two different things at the same time, but not on purpose. It is very frustrating because sometimes if someone tries to divert my attention in a different way my two thoughts merge together to make room for a new one. At most I've thought of four different things at the same time. I can actually hear them quite well. They do happen at the same time, but there is always one thought that is much louder than the others. I don't think it's normal but i know of only a few people who experience it with me, one being my father.