Have you ever played Parcheesi? I know you did. I also know that you're aware of the randomness associated with it. It's true that Parcheesi is less agonizing a game that Monopoly, just because it's objectively shorter. But that doesn't stop it from becoming even less so. Ludo Bing, which is Parcheesi at its core, also features the "Fast Ludo" mode, which is the quickest waste of time in board game history.
You can play Ludo Bing online as a guest or by supplying your Facebook credentials. Every game plays as a bet in which you wager coins. The winner gets the coins in the end. You wager coins for playing, of course, but you can also spend them on little gifts for your opponents. You can create private games by sharing a unique code with your friends.
You can also wager these coins by playing Fast Ludo. In Fast Ludo, your pawns move counterclockwise. That is the go straight into their house once they enter the board. The route is short, and there's no landing on enemy pawns.
Snake and Ladders is also available for two players. If you're not online, however, you can play Hotseat Ludo with up to four players.
The gameplay is excellent as long as you play on a big enough screen. There are some issues with selecting your pawns. Otherwise, the system tends to throw sixes more than anything else. That might be good, except that three sixes in a row will lose you your turn, and that happens rather too often as far as I'm concerned.
Ludo Bing is a nice little time waster. If you also count that you can send small presents to your opponents, you can conclude that Ludo Bing is one of the friendliest online Ludo/Parcheesi game out there.