  This paper is a follow-up to [arXiv:2001.05055] in which two-dimensional
conformal field theories in the presence of spin structures are studied. In the
present paper we define four types of CFTs, distinguished by whether they need
a spin structure or not in order to be well-defined, and whether their fields
have parity or not. The cases of spin dependence without parity, and of parity
without the need of a spin structure, have not, to our knowledge, been
investigated in detail so far.
  We analyse these theories by extending the description of CFT correlators via
three-dimensional topological field theory developed in [arXiv:hep-th/0204148]
to include parity and spin. In each of the four cases, the defining data are a
special Frobenius algebra $F$ in a suitable ribbon fusion category, such that
the Nakayama automorphism of $F$ is the identity (oriented case) or squares to
the identity (spin case). We use the TFT to define correlators in terms of $F$
and we show that these satisfy the relevant factorisation and single-valuedness
conditions.
  We allow for world sheets with boundaries and topological line defects, and
we specify the categories of boundary labels and the fusion categories of line
defect labels for each of the four types.
  The construction can be understood in terms of topological line defects as
gauging a possibly non-invertible symmetry. We analyse the case of a
$\mathbb{Z}_2$-symmetry in some detail and provide examples of all four types
of CFT, with Bershadsky-Polyakov models illustrating the two new types.
