How can I explain to my parents that I study programming languages?
If you have a few minutes, most people know how to add and multiply two three-digit numbers on paper. Ask them to do that, (or to admit that they could, if they had to) and ask them to acknowledge that they do this task methodically: if this number is greater than 9, then add a carry, and so forth. This description they just gave of what to do that is an example of an algorithm.

This is how I teach people the word algorithm, and in my experience this has been the best example. Then you can explain that one may imagine there are more complex tasks that computers must do, and that therefore there is a need for an unambiguous language to feed a computer these algorithms. So there has been a proliferation of programming languages because people express their thoughts differently, and you're researching ways to design these languages so that it is harder to make mistakes. 

This is a very recognizable situation. Most people have no concept that the computers they use run programs, or that those programs are human-written source code, or that a computer could 'read' source code, or that computation, which they associate with arithmetic, is the only thing computers do (and data movement, and networking, maybe).

My research is in quantum computing, so when people ask me what I do, I don't attempt to explain that. Instead, I try to explain that quantum physics exists (they've usually heard of Schrödinger's cat, and things that are in two places at once), and that because of this strange physics, faster computation might be possible.

My goal is to leave the person feeling a little more knowledeable than they did going in, feeling excited about a world they didn't know existed, but with which you have now familiarized them. I find that that's much more valuable than explaining my particular research questions.