Gamma-Ray Bursts (GRBs), the most luminous explosions in the cosmos, are promising tools for cosmology due to their potential as standardizable candles. Among the empirical correlations proposed for this purpose, the Yonetoku relation, which connects the intrinsic peak energy ($E_{\mathrm{i,p}}$) of the $\nu F_\nu$ spectrum to the isotropic peak luminosity ($L_{\mathrm{iso}}$), provides a mean to probe distances beyond the range of Type Ia supernovae (SNe~Ia). In this work, the Yonetoku relation is calibrated and analyzed using GRBs with measured redshifts and a large sample of GRBs with pseudo-redshifts obtained using Machine Learning (ML). This analysis focuses on estimating the distance modulus and constraining parameters of a flat $\Lambda$CDM cosmology using this relation. A joint Markov Chain Monte Carlo analysis is applied to simultaneously determine the Yonetoku relation parameters ($k, m$) and cosmological parameters ($H_{0}, \Omega_{\Lambda}$). This method is applied across the full redshift range of the GRB samples. This unified fitting strategy avoids the circularity problem in GRB cosmology, in which adopting fixed cosmological model parameters for calibrating the correlation parameters can bias subsequent cosmology parameter inference. A total of 116 {\it Fermi}-GBM GRBs with known redshifts are utilized in combination with the pseudo-redshift sample of 1576 GRBs. In addition, a combination with SNe~Ia datasets from Union~2.1, the Dark Energy Survey (DES), and Pantheon+SHOES is employed. This combined approach yields a consistent value for $H_{0}$ and $\Omega_{\Lambda}$, indicating that GRBs with well-modeled pseudo-redshifts can serve as effective high-redshift cosmological probes.