A two-dimensional frictionless adhesive contact problem for a parabolic indenter pressed against an orthotropic elastic layer resting on a smooth rigid substrate is studied in the framework of the Johnson-Kendall-Roberts (JKR) theory. In the case of a relatively small contact zone with respect to the layer thickness, the fourth-order asymptotic solution (up to terms of order) is obtained, and the pull-off force is expanded in terms of the non-dimensional measure of the work of adhesion. In particular, a pinch/compression method for soft tissue is considered, and the testing methodology is suggested based on a least-squares best fit of the first-order asymptotic model to the depth-sensing indentation data for recovering two of the three independent elastic moduli which characterize an incompressible transversely isotropic material. The case of a weakly compressible material, which is important for biological tissues, is also discussed. The developed asymptotic model can be effectively used for small values of a certain dimensionless parameter, which is proportional to the work of adhesion and the indenter radius squared, on the one side, and inversely proportional to the effective elastic modulus and the elastic layer thickness cubed, on the other.