A two-dimensional frictionless contact problem for a parabolic indenter pressed against an orthotropic elastic strip resting on a frictionless rigid substrate is studied. The sixth-order asymptotic solution is obtained in the case of a relatively small contact zone with respect to the elastic strip thickness. The so-called cylindrical lateral indentation test, which utilizes lateral contact of a cylindrical indenter, is developed for indentation testing of a thin transversely isotropic film with the symmetry plane orthogonal to the contact plane under the assumption that the film thickness is small compared to the cylinder indenter length. The presented testing methodology is based on a least squares best fit of the second-order asymptotic model to the depth-sensing indentation data. (C) 2014 Elsevier Masson SAS. All rights reserved. Document embargo 26/08/2015.