Ultimately I did what you did and realized that j+k and j-k both had to be even so after 1 and 252 I tried 2 and 126, as you suggested. But I did several other things along the way, including prime factoring 252, which led nowhere. After using difference of the squares I also tried to think of it as j plus some quantity (k) and j minus that same quantity quantity (k) both had to be factors of 252. So 18 and 14 are paired factors of 252 and so it could be 16 plus 2 and 16 minus 2. But the obviously we want j+k to be as big as possible so we want to look for factors that are spaced much farther apart. Looking back I could have pursued that line of reasoning and got to it but in the end I abandoned it and essentially did what you suggested above. I knew that j+k and j-k had to be factors of 252 so I tried them spaced far apart (1 and 252) and then solved for j and k and realized that they j+k and j-k had to be even for j and k to end up being integers. So then I tried 2 and 126 and then solved for j and k.
But that was pretty hard! I think that is a little beyond what you would see on the SAT. In my opinion people tend to really struggle with number property questions, even pretty mild ones, so they don’t have to make those kinds of questions that hard to stump most people on the test. I think that would stump almost all test takers, certainly given the time constraints.