Why did Jesus use parables to teach? At first, Jesus taught by direct instruction and expounding the scriptures. For example, Luke 2:46-47:


  ​​​46 ​And it came to pass, that after three days they found him in the ​​​temple​, sitting in the midst of the ​​​doctors​, ​​​both​ hearing them, and ​​​asking​ them questions.
  ​​​47 ​And all that heard him were astonished at his ​​​understanding​ and answers.


Also, Matthew 5:3-5:


  3 Blessed are the poor in spirit: for theirs is the kingdom of heaven.
  4 Blessed are they that mourn: for they shall be comforted.
  5 Blessed are the meek: for they shall inherit the earth.


But then, as He got closer to the end of His life, He began to teach in parables. Why?
James E. Talmage discusses this in his book &quot;Jesus the Christ&quot; (read online here)
An excerpt from this chapter:

To the chosen and devoted followers who came asking the Master why He had changed from direct exposition to parables, He explained that while it was their privilege to receive and understand the deeper truths of the gospel, “the mysteries of the kingdom of heaven” as He expressed it, with people in general, who were unreceptive and unprepared, such fulness of understanding was impossible. To the disciples who had already gladly accepted the first principles of the gospel of Christ, more should be given; while from those who had rejected the proffered boon, even what they had theretofore possessed should be taken away. “Therefore,” said He, “speak I to them in parables: because they seeing see not; and hearing they hear not, neither do they understand.”
...
There is plainly shown an element of mercy in the parabolic mode of instruction adopted by our Lord under the conditions prevailing at the time. Had He always taught in explicit declaration, such as required no interpretation, many among His hearers would have come under condemnation, inasmuch as they were too weak in faith and unprepared in heart to break the bonds of traditionalism and the prejudice engendered by sin, so as to accept and obey the saving word.
Another example of the merciful adaptation of the word of truth to the varied capacities of the people who heard the parables is found in the psychological fact, that the incidents of an impressive though simple story will live, even in minds which for the time being are incapable of comprehending any meaning beyond that of the common-place story itself. … And then, when time and experience, including suffering perhaps, had prepared them for deeper thought, they would find the living kernel of gospel truth within the husk of the simple tale.
(Talmage)

In short, listeners of varying spiritual maturity could be taught that which they needed to hear.