  We show that all lowest Landau level projected and unprojected chiral parton
type fractional quantum Hall ground and edge state trial wave functions, which
take the form of products of integer quantum Hall wave functions, can be
expressed as conformal field theory (CFT) correlation functions, where we can
associate a chiral algebra to each parton state such that the CFT defined by
the algebra is the ``smallest'' such CFT that can generate the corresponding
ground and edge state trial wave functions. A field-theoretic generalisation of
Laughlin's plasma analogy, known as generalised screening, is formulated for
these states. If this holds, we argue that the inner products of edge state
trial wave functions, for parton states where the ``densest'' trial wave
function is unique, can be expressed as matrix elements of an exponentiated
local action operator of the CFT, generalising the result of Dubail et al. [PRB
85, 11531 (2012)], which implies the equality between edge state and
entanglement level counting to state counting in the corresponding CFT. We
numerically test this result in two specific cases. We discuss how Read's
arguments [PRB 79, 045308 (2009)] still apply, implying that conformal blocks
of the CFT defined by the corresponding chiral algebra are valid quasi-hole
trial wave functions, with the adiabatic braiding statistics given by the
monodromy of these functions, assuming the existence of a quasi-particle
trapping Hamiltonian. Generalisations of these constructions are discussed. It
is shown that all chiral composite fermion wave functions can be expressed as
CFT correlation functions without explicit symmetrisation or
anti-symmetrisation and that the ground, edge, and certain quasi-hole trial
wave functions of the $\phi_n^m$ parton states can be expressed as the
conformal blocks of the $U(1) \otimes SU(n)_m$ WZW models.
