smc

Because the father's age is a two digit number, we know that the father's age must be $43$ so that the second two digits of the original number can be a number in the teens. Let the son's age be $x$. We know that the original number is $4300+x$, and the positive difference between their ages is $43-x$. Thus, we have the equation $4300+x-(43-x)=4289$, which yields $x=16$. Thus, the sum of the two ages is $43+16=\boxed{59}$.