All of you guys on this forum are the greatest. I replaced the relay with an NEC relay, and voila, my intermittent wipers work. The most important part of this is, the wipers park when you turn them off.
One bit of advice for anyone who wants to attempt this though. When you find the grey box (2002 Rodeo LSE), DO NOT attempt to take the entire box out. In the direction where the wire comes in. There's a 3 clip cover, that when removed (with your fingers), will allow you to take the entire circuit board out. A little bit of caution, de-solder and taking the old relay out is quite a pain. The NEC is a bit louder, but hey, the wipers work, and the clicking just ensures that it's working properly.
A big thank-you goes out to all you guys that helped made it possible. Just wanted to let you all know that your time to share your knowledge is very much appreciated.