In this paper, we continue our study on the cooperative periodic-parabolic system: $$\begin{aligned} \left\{ \begin{array}{ll} \begin{array}{l} \partial _tu-\Delta u=\mu u+\alpha (x,t)v-a(x,t)u^p\\ \partial _tv-\Delta v=\mu v+\beta (x,t)u-b(x,t) v^q \end{array} &{}\quad \hbox {in }\ \Omega \times (0,\infty ),\\ \; (\partial _\nu u,\partial _\nu v)=(0,0) &{}\quad \hbox {on }\ \partial \Omega \times (0,\infty ),\\ \; (u(x,0),v(x,0))=(u_0(x),v_0(x))> (0,0) &{}\quad \hbox {in }\ \Omega , \end{array}\right. \end{aligned}$$ where \(p,\,q>1, \Omega \subset {\mathbb {R}}^N\; (N\ge 2)\) is a bounded smooth domain, \(\alpha ,\,\beta >0\) and \(a,\, b\ge 0\) are smooth functions that are \(T\)-periodic in \(t\), and \(\mu \) is a varying parameter. The unknown functions \(u(x,t)\) and \(v(x,t)\) represent the densities of two cooperative species at location \(x\) and time \(t\). In [1], we dealt with the case that \(a\) and \(b\) have simultaneous temporal and spatial degeneracies (i.e., vanish), and studied the long-time behavior of \((u,v)\) which resembles that of the scalar periodic-parabolic logistic equation with temporal and spatial degeneracies. The present paper concerns the other two natural situations: simultaneous temporal degeneracy and simultaneous spatial degeneracy. When the species are exposed to such degenerate environments, our investigation reveals new dynamical behaviors, comparing to [1, 2]. The limiting behavior of the principal eigenvalue problem of an associated linear periodic-parabolic system, which may be of independent interest, plays a crucial role in our analysis.