The Historic Center of Cuenca (HCC) is located in the southern region of Ecuador. It is well known that our country is located on the so-called belt of fire of the Pacific Ocean, this area is characterized by having generated the most important seismic events in the history of mankind. More specifically, there are records that show that in the last 200 years the city of Cuenca has been exposed to earthquakes that have produced moderate to severe damage. These reasons make it possible to establish that the city of Cuenca and specifically its historic center could present important problems in the face of significant seismic events. Most of the buildings in the HCC date back to the middle of the 20th century and have used unreinforced brick masonry (brick-URM) to build their walls. This work is part of the Seismic Vulnerability Project: Seismic Damage Scenarios of the Built Heritage of the Historic Center of Cuenca. In the context of this vulnerability project, the objective of this work was to establish a family of pushover curves for three unreinforced brick masonry buildings typical of the HCC, based on a parametric pushover analysis. The definition of the typical buildings was based on an extensive work of architectural and geometric characterization of the traditional built heritage of HCC. On the basis of focusing the study on two-story buildings (the most common), the size of the floor area of the buildings (small, medium and large area) was assumed as a base parameter. Based on an analysis of the variability of different geometric and mechanical characteristics, and in order to study their influence on the pushover curves of the three typical brick URM buildings, the following study parameters were defined: 1) compressive strength of brick masonry y 2) wall thickness. The pushover analysis was carried out with the Ruaumoko program. The model of the buildings responds to an equivalent portal frame macro-model scheme that has been formulated and validated by the authors of this paper. The results will be discussed in terms of the incidence of the variability of the study parameters on the basic characteristics of the pushover curves. These results will be an essential input for the next stage of the project consisting of damage estimation for different levels of seismic action expected in the city.