An exceptional equinumerosity of lattice paths and Young tableaux

We consider families ${\mathcal{P}}_n$ of plane lattice paths enumerated by Guy, Krattenthaler, and Sagan (1992). We show by explicit bijection that these families are equinumerous with the set $\mathrm{S}\mathrm{Y}\mathrm{T}(n+2,2,1^n)$ of standard Young tableaux.