There is no way to transfer Pokemon from Gen II to Gen III, so you're out of luck. =/

Unfortunately there is no way to trade from Generation 2 to Generation 3 (My understanding is that the code between those two generations was too different.).

If you want get any Pokemon from Generation 3 though, you got to use Pal Park (I recommend using your SoulSliver as there isn't a six Pokemon a day limit in that one.), which does require you to recapture your transfer Pokemon in the Generation 4 game (The Pokeballs have 100% catch rate though.).

As for transfering to Generation 5, you have to use Poke Transfer, which basically has a catching mini-game instead of having you hunt them down to recatch them.

As for Poke Transporter, it takes entire of your Box 1 in Generation 5 games and places it in a Transport Box for moving over to Pokemon Bank storage. You will need to download Pokemon Bank and Poke Transporter from eShop to do this.

- JMJ 2014

This is actually what made people hate Gen 3 when it came out; no-one could get their awesome party from Gen 2 to Gen 3, so everyone was maddened. It's why Gamefreak was adamant about getting some form of transferring in every generation after that, I bet. =P

I could've sworn there was a "time capsule trading" or something, but I guess not.

Really appreciate the advice JMJ - at least I can get my party from Gen III onward.

The Time Capsule trading thing is only for trade between Generation 1 and 2. It also only version of Generation trading that allows you trade back to previous Generation. The rest of these Generation Trading devices all permanently transfer the Pokemon to the next Generation.

Also to added to what Blade said, this issue in a lot of ways was one of the reasons that Game Freak remade both Generation 1 and 2 in Generation 3 and 4 because it was impossible to get certain Pokemon from first two generations.

- JMJ 2014

This entry gave me feels. ;_;

You could always replace Silver's battery and recreate your party even better than before.

Well, how hard is it to replace the battery in a GBC game? I can probably live with just trading updated versions of them up from Soul Silver if I can get it into a stable enough state to figure out who to send/catch/etc.