This was my answer to your questions about doing an MA and a PhD concurrently.
I'm doing a PhD and an MFA concurrently. The process will depend on your university, but I would talk to the director of graduate studies in your home department, the graduate college at the university, and the director of graduate studies for TESL as well.
You may need to apply like any other candidate - but you have the benefit of potentially not needing TESL to outlay any money for you (your PhD is doing that already, right?), so that'll be a major point in your favor. This is also easier done if you're still relatively early in your program.
Things to find out include how much overlap you'll be allowed between your degrees - at my university, they take the PhD and look to the Master's degree from the department, compare it to the second Master's, and then take the shorter one. Then they divide the number of credits required for each by some arbitrary number (I think six), round the result down, and that's how many credits from each you'll be able to count toward the other. In my case, that number is seven, so I'll get to count seven credits of English toward Translation and seven credits of Translation toward English. All told, I'm looking at some ~32 more credits than I would need just to do the PhD. So taking courses in the summers is an absolute must.
It also means you'll have to sit down with a pen and paper and really go over your credits and plan our the rest of your path to the dual degree as far as possible, as specifically as you can.
You'll also want to make sure your Master's will feed into your dissertation in some way - I'm translating the text I'll be analyzing in my first chapter of my dissertation for my PhD, so the critical introduction to the translation will overlap quite well with that chapter.
Hope that helps.