Consider a planar Brownian motion starting at an interior point of the parabolic domain D = {(x, y) : y > x²}, and let τ_D denote the first time the Brownian motion exits from D. The tail behaviour (or equivalently, the integrability property) of τ_D is somewhat exotic since it arises from an interference of large-deviation and small-deviation events. Our main result implies that the limit of T ^-1/3 log ℙ{τ_D > T}, T → ∞, exists and equals -3π²/8, thus improving previous estimates by Bañuelos et al. and Li. The existence of the limit is proved by applying the classical Schilder large-deviation theorem. The identification of the limit leads to a variational problem, which is solved by exploiting a theorem of Biane and Yor relating different additive functionals of Bessel processes. Our result actually applies to more general parabolic domains in any (finite) dimension.