  Parametrically driven oscillators can display period-tripling in response to
a drive at thrice the resonance frequency. In contrast to the parametric
instability for period doubling, the symmetric fixed-point corresponding to the
state of rest remains stable at arbitrary strong driving for the tripling
transition. Previously, it has been shown that fluctuations can circumvent this
and induce a period-tripling instability. In this article, we explore an
alternative way of inducing a period-tripling transition by investigating
properties of period-tripling due to parametric down-conversion beyond the
rotating-wave approximation. We show that despite the absence of an instability
threshold, off-resonant frequency contributions can induce a period-tripling
transition by activating the parametric down-conversion. Moreover, we study the
subsequent period-tripled states of the Josephson potential and discuss the
asymmetry between the clockwise and counter-clockwise rotating fixed-points
that only arises beyond the rotating-wave approximation.
