We consider the parabolic Anderson model in the Stratonovich integral, with the time-$L_{loc}^{p}(\mathbb{R})$-covariance and space-$\alpha$-homogeneous Gaussian fields. We study the left-tailed probability of the solution and its density, inspired by the work of [Hu, L\^{e}, AIHP, 2022]. When $u_0$ is a nonnegative measure, the rough estimates of their upper bounds are obtained jointly owning the exponent of rate $2$. When $u_0$ is a bounded function satisfying that $\inf_{x\in\mathbb{R}^d}u_0(x)>0$, the rates of the upper bounds are improved, more precisely, the exponent of rate is $2+\frac{\alpha}{2}(1-\frac{\alpha}{4})$ for $\alpha\in(0,2)$ in the time-independent case; it is larger than $2$ for $\alpha\in(0,6-2\sqrt{5})$ in the time-dependent case.