Jesus used parables extensively in his teaching. When Jesus's disciples asked why he taught in parables he told them it was too hide the spiritual meaning from those who were not worthy, see Matthew 13:11. Although Jesus needed to protect himself from the religious leaders of his time until He was able to bring His message to the world, there are other reasons for using parables. An article on Yahoo voices gives five reasons Jesus taught with parables. 1) Jesus Used Parables to Make Difficult Principles Easier to Understand, 2) Jesus Used Parables Because They are Memorable, 3) Parables Were A Common Teaching Style During the Time of Jesus, 4) Parables Allowed Jesus to Teach Controversial Ideas Without Getting Him in Trouble with Religious Leaders of the Day, 5) Jesus Came to Show Us God Not to Teach Us Rules. I particularly like the discussion about how we all like stories. Parable and stories are something that we can think back on again and again and get more understand as we go through our own life experiences. For me the Prodigal son and the feelings of the older brother are particularly meaningful. How easy it is for pride and judging others to take us away from Jesus's teaching to love everyone. The link to Ken Collins's page on the reason for parables adds additional insight.
The Good Samaritan Luke 10:25-37. Jesus used a Samaritan as the one who showed compassion because the Jews felt of all people the Samaritans would be the least likely to be considered a neighbor. This parable has had a great influence on Western Culture. Now a Samaritan instead of being someone looked down on is know as someone that helps others.
The Prodigal Son Luke 15:11-32. This parable illustrates parental love and forgiveness. The behavior of the older brother shows how easy it is to have feeling of jealousy.
Lost sheep Luke 15:1-7. The parable of the lost sheep is similar to the parable of the Prodigal Son. Leaving the 99 in search of the one illustrates the importance God places on each individual person. A similar parable is the lost coin, Luke 15: 8-10. All these parables are in Luke 15.
Ten Virgins. Matthew 25:1-13. The parable is referred to in Luke 12:35-36 and D&C 45:56-59. In the D&C it tells us this parable refers to the Second Coming. When this parable comes up there is usually discussion about why couldn't the ones who were prepared share their oil. I think this is an example of trying to read too much into the parable. The oil represents spiritual preparedness that is not possible to share, although many a parent would dearly like to share their oil with a wayward child.
Ungrateful Servant. Matthew 18:23-35. If we expect the Lord to forgive us we need to forgive others.
Talents. Matthew 25:14-30. In this parable I believe the Lord is telling us that we are expected to develop our talents and not just sit ideally by waiting for others to act. In the Doctrine and Covenants 58:26-29 we are told to do many good works of our own free will and choice. The use of the word talent in English to describe abilities comes from this parable.
A very similar parable is of the pounds, Luke 19:11-27. In this parable each servant is given an equal number of pounds but the results are not equal. When I searched on Google for the parable of the pounds I found lots of places that called it the parable of the mina. It turns out that a mina is not much different in weight than an English pound. It was interesting to me that talents, minas, and shekels are all units of weight. They become units of money as a mina of gold, silver, etc. There are variations in the values according to time and place but it looks like 1 talent = 60 mina and 1 mina = 50 shekels.
The parable of the pounds also has the part about receiving a kingdom and slaying his enemies. I have wondered why those parts of the parable were included unless maybe if was for the benefit of the people in Jesus' time. It sets the background of the parable since it matches the events that had happened just a few years earlier with Archelaus son of Herod.
Sower and the seed. Matthew 13:3-9, Mark 4:3-9 and Luke 8:4-8. Jesus interprets the parables to his disciples Matthew 13:18-23, Mark 4:14-20 and Luke 8:11-15. This parable is not as well know as parables like the Good Samaritan or the the Prodigal Son but it is found in all the synoptic gospels. When I was on a mission to Brazil this parable was a comfort. Some of the people that I baptize didn't stay active in the Church. I worried that I hadn't taught them correctly. However, as this parable show it has always been that some people start out strong, but then fall away. As missionaries the writers of the gospels probably also identified with this parable and so included it in their writings.
Mustard seed Matthew 13:31-32, Mark 4:30-32, Luke 13:18-19. The parable of the mustard seed is very simple but unlike some of Jesus's more well know parables is found in all the synoptic gospels. I wonder if it is because the main work of the Apostles was to make the kingdom grow.
laborers in The Vineyard Matthew 20:1-16. At times it can be easy to be jealous of those who turn around their life later in life and who still receive the full blessings of the Gospel.
Candle Under a Bushel Matthew 5:14-16. This is the scripture that tells us that we need to let our good works be seen before men so that they will glorify our Father in Heaven. In the LDS Harmony of the gospels this is entitled "Ye are the light of the world. There is also an entry for candle with scriptures: Mark 4:21-25 and Luke 8:16-18. These scriptures also talk about putting a candle under a bushel or under a bed, but the relationship to missionary work is less clear.
The great supper, Luke 14:12-24. This is the parable where a man made a supper and none of the invited guests came. Instead the poor and humble people were invited. This parable shows that in the Kingdom of God it isn't the rich and famous that have priority as it is here on earth but instead the humble.
Wedding of a king's son Matthew 22:1-14. This parable is similar to the great supper where the invited guests refuse to come so the poor and the humble are invited instead. One difference is the man there without a wedding garment. Some people have speculated that the wedding garments were provided by the host. I think the significance of this incident is that everyone that enters the kingdom of God must abide by the rules that God as set up. There are no exceptions.
Lazarus and the rich man, Luke 16:19-31. To my knowledge no other person in Christs parables were given names. Here the poor man is named Lazarus foretelling that Jesus would raise a different Lazarus from the dead. Just as the parable predicted in verse 31 people were not to "be persuaded, though one rose from the dead."
Unjust steward Luke 16:1-13. This parable is a bit strange because it seems to look favorably on dishonestly. However, in verse 13 it says "Ye cannot serve God and mammon" warning us about searching for riches.
I like trying to come up with parables or allegories that relate to the modern world or the world of science.