Parent-identifying schemes provide a way to identify causes from effects for some information systems, such as digital fingerprinting and group testing. In this paper, we consider the combinatorial structures for parent-identifying schemes. First, we establish an equivalent relationship between the parent-identifying schemes and forbidden configurations. Based on this relationship, we derive the probabilistic existence lower bounds for two related combinatorial structures, that is, <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-parent-identifying set systems (<inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-IPPS) and <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-multimedia parent-identifying codes (<inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-MIPPC), which are used in broadcast encryption and multimedia fingerprinting, respectively. The probabilistic lower bound for the maximum size of a <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-IPPS has the asymptotically optimal order of magnitude in many cases, and that for <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-MIPPC provides the asymptotically optimal code rate when <inline-formula> <tex-math notation="LaTeX">t=2 </tex-math></inline-formula> and the best known asymptotic code rate when <inline-formula> <tex-math notation="LaTeX">t\ge 3 </tex-math></inline-formula>. Furthermore, we analyze the structure of 2-IPPS and prove some bounds for certain cases.