  An association scheme is called partially metric if it has a connected
relation whose distance-two relation is also a relation of the scheme. In this
paper we determine the symmetric partially metric association schemes with a
multiplicity three. Besides the association schemes related to regular complete
$4$-partite graphs, we obtain the association schemes related to the Platonic
solids, the bipartite double scheme of the dodecahedron, and three association
schemes that are related to well-known $2$-arc-transitive covers of the cube:
the M\"{o}bius-Kantor graph, the Nauru graph, and the Foster graph F048A. In
order to obtain this result, we also determine the symmetric association
schemes with a multiplicity three and a connected relation with valency three.
Moreover, we construct an infinite family of cubic arc-transitive
$2$-walk-regular graphs with an eigenvalue with multiplicity three that give
rise to non-commutative association schemes with a symmetric relation of
valency three and an eigenvalue with multiplicity three.
