What about the Rainhill contest?

Steam Engine Grad Prix (Budapest- Hungary)

http://http://www.youtube.com/watch?feature=player_detailpage&v=1AS6RszQGA4

http://www.youtube.com/watch?feature=player_detailpage&v=1AS6RszQGA4

Commuter trains from Madrid

JR Chūō Line Rapid VS JR Keihin-Tōhoku Line VS JR Yamanote Line (Kanda, Tōkyō)

JR Yamanote Line VS JR Keihin-Tōhoku Line VS JR Tōkaidō Shinkansen
(Shinagawa - Ōsaki, Tōkyō)

Keikyū Main Line VS JR Yokosuka Line
(Keikyū-Kawasaki/Kawasaki - Yokohama, Kanagawa)

Hankyū Kōbe Line VS Hankyū Takarazuka Line VS Hankyū Kyōto Line
(Umeda - Jūsō, Ōsaka)

Sanyō Railway Main Line VS JR Kōbe Line (New Rapid) VS JR Kōbe Line (Rapid)
(Maiko-Kōen - Akashi, Hyōgo)

Countless more from Japan can be found by searching "並走" (heisō, parallel run).

A train race in Serbia:

HST vs Voyager!

Paris Saint Lazare suburban network

International trains Amsterdam-Brussels running parallel between Leiden and the Hague.

Part 1:
Part 2:

that's actually cool

JR Shōnan-Shinjuku Line vs. JR Takasaki Line vs. JR Keihin-Tōhoku Line
Akabane → Minami-Urawa

Source: 1947FGALL on YouTube

Even better battle taken from the opposite side, inside a Takasaki Line train.
Akabane → Urawa

Source: 1947FGALL on YouTube

Tōkyū Meguro Line vs. Tōkyū Tōyoko Line
Tamagawa → Musashi Kosugi

Source: SakuranoKurimu8642 on YouTube

Nankai Kōya Line vs. Nankai Main Line
Shin-Imamiya → Namba

Source: Meitetsu6750kei on YouTube

Omg I never heard abotu this
funny videos
I didnt know it exist!