We present a routine for 3D magnetotelluric (MT) modeling based upon high-order edge finite element method (HEFEM), tailored and unstructured tetrahedral meshes, and high-performance computing (HPC). This imple- mentation extends the PETGEM modeller capabilities, initially developed for active-source electromagnetic methods in frequency-domain. We assess the accuracy, robustness, and performance of the code using a set of reference models developed by the MT community in well-known reported workshops. The scale and geological properties of these 3D MT setups are challenging, making them ideal for addressing a rigorous validation. Our numerical assessment proves that this new algorithm can produce the expected solutions for arbitrarily 3D MT models. Also, our extensive experimental results reveal four main insights: (1) high-order discretizations in conjunction with tailored meshes can offer excellent accuracy; (2) a rigorous mesh design based on the skin-depth principle can be beneficial for the solution of the 3D MT problem in terms of numerical accuracy and run-time; (3) high-order polynomial basis functions achieve better speed-up and parallel efficiency ratios than low-order polynomial basis functions on cutting-edge HPC platforms; (4) a triple helix approach based on HEFEM, tailored meshes, and HPC can be extremely competitive for the solution of realistic and complex 3D MT models and geophysical electromagnetics in general.