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 Hubble parameter (H) is considered for the purpose in the flat Friedmann–Lemaitre–Robertson–Walker 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 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 datasets. With the constrained values of the model parameters, both the models are analyzed and compared with the standard ΛCDM model and showing nice fit to the datasets. Two different candidates of DE are 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 detail, and the evolution of various cosmological parameters is shown graphically. The age of the Universe in both models is 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. © 2020, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.