On studying Computer Science vs. Software Engineering to become a proficient coder I’m mostly interested in the coding aspect of the CS and SE majors curricula. And I’m trying to decide between the two. 

Each major has its pros and cons, and they prepare you in very different ways for a career in coding.

Computer science courses that are provided at my local university range from parallel programming to algorithm design. There's also computer organization/architecture and web programming.

It is to my understanding that (some of) these are exclusive to CS.

I know this post might attract subjective opinions but I'm not looking for answers that simply fill in the blanks. Instead I hope for a final answer where perhaps one major is preferred to the other. Is a question of this nature feasible?
You can find a fair amount of information about accredited programs in the US here, including links to schools that have one or the other programs. 

In general, Software Engineering is more directed at the processes involved in creating software, and Computer Science is more generally focused on the underlying theory.

But, I'd suggest that both programs have a focus beyond just programming. In fact, you don't need any formal education to be a programmer, so either sort of program will help you develop programming skills. But that isn't the actual main focus of either. For example, an individual programmer may have little need to know about large scale software process so long as they can do their part in the larger scheme of things. He or she may be able to build things while having little understanding of the theory.

I'll also say that a "career" as a coder isn't going to be especially rewarding, either in pay or in responsibility. Many people start out there, but it is much more important, in any field, to know what should be built rather than how to build things. A degree program will give you some insight into the more important problem (though more is needed). But an entry level job after a degree may require programming but if it is the end point then your education was really wasted.

I, myself, sometimes do a lot of coding, but it is never the main thing. It is just a tool that can be used for the realization of something important. Focus on that last part, not the actual coding.



Note that Tim Cook of Apple has recently said that you don't need a degree to be a coder. Actually, though, he meant more than just "coder".