The time-dependent-asymmetric-linear parsimony is an ancestral state reconstruction method which extends the standard linear parsimony (a.k.a. Wagner parsimony) approach by taking into account both branch lengths and asymmetric evolutionary costs for reconstructing quantitative characters (asymmetric costs amount to assuming an evolutionary trend toward the direction with the lowest cost). A formal study of the influence of the asymmetry parameter shows that the time-dependent-asymmetric-linear parsimony infers states which are all taken among the known states, except for some degenerate cases corresponding to special values of the asymmetry parameter. This remarkable property holds in particular for the Wagner parsimony. This study leads to a polynomial algorithm which determines, and provides a compact representation of, the parametric reconstruction of a phylogenetic tree, that is for all the unknown nodes, the set of all the possible reconstructed states associated to the asymmetry parameters leading to them. The time-dependent-asymmetric-linear parsimony is finally illustrated with the parametric reconstruction of the body size of cetaceans.