This work explores the application of sparse identification techniques coupled with autoencoder neural networks for model order reduction of bifurcating phenomena in computational fluid dynamics. We extend previous studies, by offering novel perspectives to enhance computational efficiency and interpretability for non-uniqueness behaviors in fluid dynamics systems, focusing on two test cases: a pitchfork bifurcation caused by the Coandă effect in a sudden- expansion channel, and a Hopf bifurcation in a contraction-expansion channel. Our approach combines the Sparse Identification of Nonlinear Dynamics (SINDy) method with autoencoders and proper orthogonal decomposition (POD) to create interpretable reduced-order models capable of capturing complex nonlinear evolutions. This SINDy-AE-POD architecture allows for efficient computations, providing insights into the underlying physical processes, and learning low-dimensional representations of the governing equations. We demonstrate the ability of this approach to reconstruct full-order solutions accurately, extrapolate for unseen parameter values, and recover bifurcation diagrams. Results show that the methodology effectively captures the symmetry-breaking pitchfork bifurcation and the onset of periodic solutions in the Hopf bifurcation case. The reduced models achieve significant computational speedup while maintaining accuracy in predicting system behavior across bifurcation regimes.