  The concepts of symmetric and asymmetric peaks in Dyck paths were introduced
by Fl\'{o}rez and Ram\'{\i}rez, who counted the total number of such peaks over
all Dyck paths of a given length. Elizalde generalized their results by giving
multivariate generating functions that keep track of the number of symmetric
peaks and the number of asymmetric peaks. Elizalde also considered the
analogous notion of symmetric valleys by a continued fraction method. In this
paper, mainly by bijective methods, we devote to enumerating the statistics
"symmetric peaks", "asymmetric peaks", "symmetric valleys" and "asymmetric
valleys" of weight $k+1$ over all (partial) Dyck paths of a given length. Our
results refine some consequences of Fl\'{o}rez and Ram\'{\i}rez, and Elizalde.
