  We use the manifestly conformally invariant description of a Lorentzian
conformal structure in terms of a parabolic Cartan geometry in order to
introduce a superalgebra structure on the space of twistor spinors and normal
conformal vector fields formulated in purely algebraic terms on parallel
sections in tractor bundles. Via a fixed metric in the conformal class, one
reproduces a conformal superalgebra structure which has been considered in the
literature before. The tractor approach, however, makes clear that the failure
of this object to be a Lie superalgebra in certain cases is due to purely
algebraic identities on the spinor module and to special properties of the
conformal holonomy representation. Moreover, it naturally generalizes to higher
signatures. This yields new formulas for constructing new twistor spinors and
higher order normal conformal Killing forms out of existing ones, generalizing
the well-known spinorial Lie derivative. Moreover, we derive restrictions on
the possible dimension of the space of twistor spinors in any metric signature.
