In Act I, scene ii, we learn that Paris is interested in marrying Juliet, and that Capulet warms him that it will not be wholly his choice, but that Juliet herself will have a say -- a very unusual position for a father to take in Shakespeare's day. Capulet does invite Paris to his party:
This night I hold an old accustom'd feast
Whereto I have invited many a guest
Such as I love, and you among the store. . .
He actually goes on to encourage Paris to look over all the girls who will be there to "like her most whose merit most shall be," cautioning Paris that Juliet is but "one" of many young ladies who will attend, indicating that Capulet is not so sure that this match is a good one.
In the party scene itself (Act I, scene v), Paris does not have any lines, but you should be careful not to assume that this means he is not onstage. Shakespeare, like any playwright, understood that the staging of any scene involves much more than just the speaking of lines written. Consequently, Paris' appearance at the party and his potential interaction with Juliet is up to those staging the play. Juliet is actually talked about by Romeo before they meet. Does he, for example, describe her as she is in private conversation with Paris? This sort of decision, again, is up to those staging the play.
Paris, then, is very likely at the party scene, and also, potentially, actually interacting with Juliet. All of this is not scripted by Shakespeare, however, and is meant to be constructed during the staging of the play.
It is also worth noting that Romeo attends the party because he sees Rosaline's name on the list which Capulet's servant has him read earlier in Act I. She is also not a "speaking" character in this scene, but her presence onstage just as likely as that of Paris.
For more on Act I, scene v, please follow the links below.