Gary/ your rival's Eevee doesn't evolve based on your Pikachu's happiness.
>The rival will eventually evolve his Eevee (and thus affect which are the other members of his party) depending on the player's actions early in the game:
>- If the player beats the rival at both the Oak Pokémon Research Laboratory and the early optional battle at Route 22, Eevee evolves into Jolteon.
- If the player loses one of these two battles or passes up the Route 22 battle, Eevee evolves into Flareon.
- If the player loses at Oak's Lab, and loses or passes up the Route 22 battle, Eevee evolves into Vaporeon.
I guess your Pikachu's happiness is linked, as winning battles will make it more happy, but that in itself doesn't determine your rival's Eeveelution.
Pokemon Yellow - Bulbapedia