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The presented paper is a comprehensive analysis of two dark energy (DE) cosmological models wherein exact solutions of the Einstein field equations (EFEs) are obtained in a model-independent way (or by cosmological parametrization). A simple parametrization of the Hubble parameter ($H$) is considered for the purpose in the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background. The parametrization of $H$ covers some known models for some specific values of the model parameters involved. Two models are of special interest which show the behavior of cosmological phase transition from the deceleration in the past to acceleration at late-times. The model parameters are constrained with $57$ points of Hubble datasets together with the $580$ points of Union $2.1$ compilation supernovae datasets and baryonic acoustic oscillation (BAO) datasets. With the constrained values of the model parameters, both the models are analyzed and compared with the standard $Λ$CDM model and showing nice fit to the datasets. Two different candidates of DE is considered, cosmological constant $% Λ$ and a general scalar field $ϕ$ and their dynamics are discussed on the geometrical base built. The geometrical and physical interpretations of the two models in consideration are discussed in details and the evolution of various cosmological parameters are shown graphically. The age of the Universe in both models are also calculated. Various cosmological parametrization schemes used in the past few decades to find exact solutions of the EFEs are also summarized at the end which can serve as a unified reference for the readers.