Elliptic curves with pairings at 128-bit security in libpbc?
The security of pairing-based cryptography relies on the security of the elliptic curve (which is linked to the size of underlying finite field, or "base field") and of the finite extension field being used.

The "Dlog security" column in the linked page is the size of the finite extension field. Its security used to be comparable to the corresponding RSA security; 1024 bits would provide only 80 bits of security. However, The "Dlog security" of the extension fields used in pairings has changed with some recent research, which makes things even more complicated. See Updating key size estimations for pairings, which estimates, for example, that a BN curve with a base field of 451 bits and "Dlog security" of 5532 bits would provide 128 bits of security)

The "Base field size" is related to the elliptic curve security; in this case security is usually half of the field size (to be precise: half the size of the prime-order subgroup). The "type A" curve there would provide 256 bits of security; but since the corresponding Dlog security is only 80 bits, the overall security is only 80.

Considering that, none of the listed benchmarked curves in the page you linked to provide 128-bit security. However, libpbc does support generating curves with arbitrary sizes for some pairing types, so it is possible to generate a pairing with 128-bit security.