I didn’t really consider what the parser might be looking for, but I would consider that another ‘user’ you have to design for. That leaves two different options: design in a way that satisfies both or have two different resumes. I have seen enough ‘traditional’ resumes (not through a parser) in both corporate and studio that they are still very much important. Additionally, they are useful when networking and meeting people in different formats. I have a feeling that if the ATS software and design managers look for different things then the ATS software needs to be reconsidered.
As for keeping things to one page — when I work on my resume, I am constantly evaluating what goes in and what goes out. There is a lot more I could be describing that I don’t. I put what I think will have the maximal impact and what aligns best with the job description. It is not easy, but neither is being a designer.