Modern high-pressure turbine (HPT) blade design stands out due to its high complexity comprising three-dimensional blade features, multipassage cooling system (MPCS), and film cooling to allow for progressive thermodynamic process parameters. During the last decade, probabilistic design approaches have become increasingly important in turbomachinery to incorporate uncertainties such as geometric variations caused by manufacturing scatter and deterioration. Within this scope, the first part of this two-part article introduces parametric models for cooled turbine blades that enable probabilistic finite element (FE) analysis taking geometric variability into account to aim at sensitivity and robustness evaluation. The statistical database is represented by a population of more than 400 blades whose external geometry is captured by optical measurement techniques and 34 blades that are digitized by computed tomography (CT) to record the internal geometry and the associated variability, respectively. Based on these data, parametric models for airfoil, profiled endwall (PEW), wedge surface (WSF), and MPCS are presented. The parametric airfoil model that is based on the traditional profile theory is briefly described. In this regard, a methodology is presented that enables to adapt this airfoil model to a given population of blades by means of Monte Carlo-based optimization. The endwall variability of hub and shroud are parametrized by radial offsets that are applied to the respective median endwall geometry. WSFs are analytically represented by planes. Variations of the MPCS are quantified based on the radial distribution of cooling passage centroids. Thus, an individual MPCS can be replicated by applying adapted displacement functions to the core passage centroids. For each feature that is considered within this study, the accuracy of the parametric model is discussed with respect to the variability that is present in the investigated blade population and the measurement uncertainty. Within the scope of the second part of this article, the parametric models are used for a comprehensive statistical analysis to reveal the parameter correlation structure and probability density functions (PDFs). This is required for the subsequent probabilistic finite element analysis involving real geometry effects.