  Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard
Brownian motion marginals and constant correlation $\rho \in (-1,1).$ Parisian
ruin is defined as a classical ruin that happens over an extended period of
time, the so-called time-in-red. We derive exact asymptotics for the
non-simultaneous Parisian ruin of the company conditioned on the event of
non-simultaneous ruin happening. We are interested in finding asymptotics of
such problem as $u \to \infty$ and with the length of time-in-red being of
order $\frac{1}{u^2},$ where $u$ represents initial capital for the companies.
Approximation of this problem is of interest for the analysis of Parisian ruin
probability in bivariate Brownian risk model, which is a standard way of
defining prolonged ruin models in the financial markets.
