We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e., the estimation error), whereas the second is more specifically oriented to the moments of the estimation error. The proposed bounds are relatively easy to calculate numerically (in the sense that their optimization is over relatively few auxiliary parameters), yet they turn out to be tighter (sometimes significantly so) than previously reported bounds that are associated with similar calculation efforts, across a variety of application examples. In addition to their relative simplicity, they also have the following advantages: (i) Essentially no regularity conditions are required regarding the parametric family of distributions; (ii) The bounds are local (in a sense to be specified); (iii) The bounds provide the correct order of decay as functions of the number of observations, at least in all examples examined; (iv) At least the first family of bounds extends straightforwardly to vector parameters.