Literature on the DSA (e.g. Gupta 2014) agrees that the security in DSA relies on the difficulty to solve two distinct discrete logarithm problems (DLP):
One is the DLP in prime-order ($q$) subgroups of $\mathbb{Z}_p^*$ for which the most efficient algorithm is Pollard's rho method. I understand this is basically calculating the private key $x$ given the corresponding public key $y$ for $y=g^x$ mod $p$, and to avoid this we choose a large enough $q$ (e.g. 160-bit).
The other DLP is said to be in $\mathbb{Z}_p^*$ itself, to which the Number Field Sieve algorithm would apply; but I am uncertain what the meaning behind this second DLP is, i.e. what is the exact calculation that we try to do here, and how would it affect the security of digital signatures that are based on subgroups, not the overall $\mathbb{Z}_p^*$?
My initial idea was that it is because the domain parameters are in $\mathbb{Z}_p^*$, but I can't find online sources or literature that goes into more detail regarding this point.
What is the Discrete Logarithm Problem in (Z/Zp) in the DSA?