In Romeo and Juliet, do Juliet's parents force her to marry Paris?

Because of her social position and historical context, Juliet is representative of a girl who has no real voice in many decisions which govern her life. Her marriage to Paris is arranged with her father and without any of her own input. In act 1, scene 2, we learn that Paris has requested to marry Juliet that and Lord Capulet has some hesitations. Juliet "hath not seen the change of fourteen years" (I.ii.9), and Lord Capulet believes that marriage might force Juliet to grow up too quickly. But because Paris represents a good match for the Capulet family, he eventually persuades Juliet's father to agree to the marriage.

Meanwhile, Juliet falls in love with Romeo. She has no plans to marry Paris and is fairly shocked when the wedding date is set with only three days of notice.

In act 3, scene 5, all of this comes to a climax when Lord Capulet demands that Juliet do as commanded and marry Paris. Juliet begs to avoid this hasty marriage, noting that Paris has not even come "to woo" her.

Lord Capulet explodes. Juliet has crossed several lines in her refusal to marry Paris. First, as a woman, she is expected to submit to the desires of male leadership. Second, as a daughter, she is expected to submit to her father.

Juliet's attempts to forge her own path in life and to make her own choice in marriage meets a harsh response from her father, who tells her,

Thursday is near. Lay hand on heart, advise.
An you be mine, I’ll give you to my friend.
An you be not, hang, beg, starve, die in the streets,
For, by my soul, I’ll ne’er acknowledge thee. (III.v.201-204)

Though Juliet fakes her own death to avoid the marriage to Paris, her father fully intends to force her into the marriage against her wishes—because her wishes mean nothing to him in their patriarchal society.

In Romeo and Juliet, do Juliet's parents force her to marry Paris?

When Paris first says he wants to marry Juliet, Lord Capulet (her father) specifically says "woo her" and indicates that his permission is only part of the deal and that Juliet has to agree too.

Later, Paris asks again and Lord Capulet says Juliet should listen to his suggestion and marry Paris. That's in between.

Eventually, though, it does come to force: Lord Capulet says he will drag her to the church to get married.

In Romeo and Juliet, why does Juliet's family want her to marry Paris and not Romeo?

From the Capulet's point of view there are good reasons why they want Juliet to marry Paris and not Romeo. The principal reason is that they know absolutely nothing about the relationship between Romeo and Juliet. In an example of dramatic irony, only the audience and three other characters (Friar Lawrence, the Nurse and Balthasar) are aware that Romeo has married Juliet. Had they known, the Capulets probably would have attempted to have the marriage ended. After all, Romeo was a member of a family which had been at odds with the Capulets for many years and the feud between the two families often caused violence in the streets of Verona. Another reason they may have preferred Paris was because he was related to the Prince and that would have been a political advantage for Lord Capulet, though this is only a guess, because Shakespeare never offers much information about Paris, other than he is a Count (suggesting nobility) and a "kinsman" of Prince Escalus. Even though Lord Capulet speaks well of Romeo in Act I, Scene 5, it is doubtful he would have approved a marriage between a Montague and his daughter. His angry outburst at the end of Act III shows that he could be quite temperamental and news of a secret marriage might have caused him to become even more enraged. 

Why do Juliet's parents arrange her marriage to Paris?

In part, the answer is because arranged marriages between the elite classes were the norm.  In arranged marriages, the thinking was to create political and social allegiances.  Lands and monies could be drawn together between amicable families, thus increasing the power of both families.  Love marriages, at the time of Shakespeare's writing, were just coming into vogue. 

Shakespeare appears to be considering the consequences of the new movement and the possible re-alignment of patriarchy, but does so by safely removing the conflict to another part of the world.  “Let’s see what happens when the old norms are removed, when people in authority do not follow the proper rules, and when young people subvert the system,” Shakespeare seems to be saying.  Looked at in this light, the play is less a romantic plea for choice than a warning to the populace about the dangers of upsetting the system and the problems caused by players who do not follow the established rules. 

However, we do see the influence of the shift toward love marriages.  Lord Capulet tells Paris, "Her will to my consent is but a part, / And she agreed, with in the scope of her choice"  (1.2.16-17).  Here we see some capitulation to love marriages, but a very limited choice.  Juliet is free to choose, ostensibly, so long as her selection meets with her father's approval.