  We obtain new results concerning Lang-Trotter conjecture on Frobenius traces
and Frobenius fields over single and double parametric families of elliptic
curves. We also obtain similar results with respect to the Sato-Tate
conjecture. In particular, we improve a result of A.C. Cojocaru and the second
author (2008) towards the Lang-Trotter conjecture on average for polynomially
parameterized families of elliptic curves when the parameter runs through a set
of rational numbers of bounded height. Some of the families we consider are
much thinner than the ones previously studied.
