It is an open problem whether weak bisimilarity is decidable for Basic Process Algebra (BPA) and Basic Parallel Processes (BPP). A PSPACE lower bound for BPA and NP lower bound for BPP were demonstrated by Stribrna. Mayr recently achieved a result, saying that weak bisimilarity for BPP is $\Pi_2^P$-hard. We improve this lower bound to PSPACE, and, moreover, prove this result for the restricted class of normed BPP. It is also not known whether weak regularity (finiteness) of BPA and BPP is decidable. In the case of BPP there is a $\Pi_2^P$-hardness result by Mayr, which we improve to PSPACE. No lower bound has previously been established for BPA. We demonstrate DP-hardness, which, in particular, implies both NP and co-NP-hardness. In each of the bisimulation/regularity problems we also consider the classes of normed processes. Finally, we show how the technique for proving co-NP lower bound for weak bisimilarity of BPA can be applied to strong bisimilarity of BPP.