Asking parents to use flashcards at home to help their children master the facts is probably useless, as it will be drudgery for both parent and child. I know some students have been helped by those multiplication table songs. Perhaps you could send a tape (cheap to make a copy) home with a child and ask parents to see that he listens to it nightly. I teach fifth grade and have found that games are best for those who have trouble memorizing facts. There are many games using playing cards and dice. Here are a few I use:
#1-This is played by two people with a deck of cards with the jokers and face cards removed (or jack = 11 and queen = 12). Shuffle the deck and deal them all out face down. Each player flips over a card from his or her pile. The first player to call out the correct answer gets to collect two flipped over cards. If a player calls out the wrong answer the other player gets the cards. Players continue until all the cards have been flipped over. The winner is the player with the most cards at the end. This could be adapted so the parent does not compete to call out answers, but does get to keep the cards of incorrect answers by his/her child.
#2- Another card game is like the game "War". Each player (there can be more than two) flips two cards and has to call out the product of their two cards. The player whose cards make the highest product wins all four cards. If a player calls out an incorrect product, he loses his cards to the other player (whether his product was higher or not). If the two products are the same, they leave the cards out and flip two more until someone wins all of those cards. Play can continue until one player loses all his cards, but I usually just put a time limit on the game and the player with the most cards wins.
Dice work well because we usually teach the tables up to twelves and the sum of two dice is twelve.
(You can use variations of the card games, but give points instead.)
#1- Two or more players play with four dice. Players take turns rolling two dice (a two and a four would be a six), then the other two dice (a five and a six would be eleven). Then they multiply the two numbers (six times eleven) to find the product. If the product is 50 or greater, they get a point, if less than 50... no point. If they give the incorrect product, they lose a point. Play continues for a set amount of time. The player with the most points wins! My students love this game and it's great to listen to them. For example, they will roll a seven with the first two dice and I'll hear them saying, "Oh, I need at least an eight to get a point!". Also, I think they just like to roll dice.
Perhaps you could send card and dice game ideas home to parents and ask that they play them with their kids.