  In this paper, we find the higher order expansion parameters $\alpha$ and
$\lambda$ of spherically symmetric parametrized Rezzolla--Zhidenko (PRZ)
spacetime by using its functions of the radial coordinate. We subject the
parameters of this spacetime to classical tests including weak gravitational
field effects in Solar System, observations of the S2 star located in the star
cluster close to the Sgr A$^{\star}$ and of the frequencies of selected
microquasars. Based on this spherically symmetric spacetime we perform the
analytic calculations for Solar System effects like perihelion shift, light
deflection, and gravitational time delay so as to determine limits on the
parameters by using observational data. We restrict our attention to the limits
on the two higher order expansion parameters $\alpha$ and $\lambda$ that
survive at the horizon or near the horizon of spherically symmetric metrics.
The properties of these two small parameters expansion in PRZ parametrization
are discussed. We further apply the Monte Carlo Markov Chain (MCMC) simulations
to analyze and obtain the limits on the expansion parameters by using
observations of phenomena of the S2 star. Finally, we consider the epicyclic
motions and derive analytic expressions of the epicyclic frequencies. Applying
these expressions to the quasiperiodic oscillations (QPOs) of selected
microquasars allows us to set further limits on the parameters of the PRZ
spacetime. Our results demonstrate that the higher order expansion parameters
can be given in the range $\alpha\, ,\lambda=(-0.09\, , 0.09)$ and of order
$\sim 10^{-2}$ as a consequence of three various tests and observations.
