We prove that a translation surface that has two transverse parabolic elements has a totally real trace field. As a corollary, nontrivial Veech groups that have no parabolic elements do exist.
The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms
"Veech groups without parabolic elements." Duke Math. J. 133 (2) 335 - 346, 1 June 2006. https://doi.org/10.1215/S0012-7094-06-13326-4