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Phase transitions are among the most intriguing phenomena in physical systems, yet the physics near criticality remain challenging to study using classical algorithms. Parameterized quantum circuits (PQCs) offer a promising approach to investigating such regimes on practical quantum computers. However, in order to use it to probe critical behavior, a PQC itself should be non-trivial and exhibit a phase transition and non-analyticity--a property that has not yet been clearly identified. In this work, we identify a mechanism for generating non-analyticities in PQCs. As a concrete realization, we construct a class of sequential PQCs whose observable expectation value is a non-analytic function of the circuit parameter in the infinite volume limit, showing that the prepared PQC states undergo a phase transition at the non-analytic points. The entanglement and the identified order parameter have distinct behaviors in different phases, revealing the phase diagram of the PQC state. We show that classical simulation of this PQC based on tensor networks and Pauli propagation gets less efficient in the vicinity of the phase transition point, indicating a route towards practical quantum advantage using PQCs with phase transitions.