There are many areas of science in which Euler can be called a founding father, though in most cases some predecessors can be found:
In mathematics and mechanics: Calculus of variations (Newton and Bernoulli were predecessors), fluid dynamics (again Newton and Bernoulli can be counted as predecessors), rotation of a rigid body, elliptic integrals, use of generating functions in number theory, the theory of partitions, elasticity theory, complex function theory. He also discovered Gamma, Beta, zeta and hypergeometric functions,
and possibly Bessel functions, which qualifies him as the father of the Special function theory. (I do not even mention that he discovered the exponential function of a complex argument too). Euler is also a founding father of holomorphic dynamics which is very fashionable nowadays: he was the first to
study systematically the iteration of the exponential functions (though here too he had a predecessor, marquis de Condorcet). 90% of the modern undergraduate course of Ordinary Differential Equations is also due to Euler.
In physics and engineering: interior ballistics (no predecessors that I know),
perturbation theory in celestial mechanics (predecessor: Newton).