In this thesis, we study different approaches for solving the Schrödinger equation for quantum parametric oscillators. The Wei-Norman algebraic approach, the Lewis- Riesenfeld invariant approach, the Malkin-Manko-Trifonov approach are investigated. For each approach, the wave function solutions of the Schrödinger equation, the propagator and dynamical invariants are found and their relations with each other are shown. In the Wei-Norman Algebraic approach, for constructing wave functions, explicit form of evolution operator is obtained uniquely in terms of two linearly independent classical solutions of the corresponding classical equation of motion. In Lewis-Riesenfeld approach, quadratic invariants are found in terms of the solution of Ermakov-Pinney equation and using the eigenstates of these invariants, wave function solutions are constructed. Setting initial values for Ermakov-Pinney solution, results of Wei-Norman and Lewis- Riesenfeld approaches are compared, then this solution is expressed in terms of same two linearly independent classical solutions. In Malkin-Manko-Trifonov approach, linear invariants which are symmetry operators for the Schrödinger equation, are constructed in terms of complex-valued solutions of the classical equation. Using these invariants, quadratic invariants are constructed and their eigenstates are used to find wave function solutions. Moreover, initial values for complex solutions of classical equation of motion are posed, and comparison of the three approaches is given.