Is there a general formula for three Pythagorean Triangles which share an area?
COMMENT.-Basically your problem is to find many positive integer solutions of the equation $$xy(x^2-y^2)=k$$ where $k$ is a natural integer (the area of a right triangle).
The given parametrization (from Enrique Zeleny) gives an infinity of sets of three triangles sharing the same area. In the case of Gerry Myerson's example corresponding to the equation $$xy(x^2-y^2)=341880$$ the value of $k$ is something large and is not given a similar parametrization for sets of four triangles. I try here to explain this last fact.
The curve $xy(x^2-y^2)=k$ has no singular points so it have genus equal to $\dfrac{(4-1)(4-2)}{2}=3$. We are facing a quartic plane curve of genus $3$ so by the celebrated Faltings's theorem proving the Mordell Conjecture this curve has only a finite number of rational points (nothing in this context about integer points which is a more difficult problem). I want to say that the posed problem is quite difficult. Just for sets of four triangles, certainly that $k=341880$ must be a kind of minimum possible therefore for sets of five,six,etc triangles the problem becomes even more difficult.And extremely difficult or impossible a parametrization like for three triangles.
The difficulty in the determination of integer points in a curve of genus $1$ can be indirectly visualized by reading the article "Integer points on Curves of Genus 1" written by Joseph H. Silverman: Journal of the London Mathematical Society, Volume s2-28, Issue 1, August 1983, p.1-7. So much more then for genus greater than $1$.