I cannot think of a proof or disproof at the moment but in 'Path factors in cubic graphs', Kawarabayashi, Matsuda and Oda proved that every 2-connected cubic graph of order at least six has a {$P_3$,$P_4$}-factor. (Actually, they showed that every such graph has a {$P_k$| $k$≥6}-factor.) I am not sure you can renounce the implicit minimum degree condition and still hope for the same conclusion.