Day, night, month, year! My constant care has been to have my
only child worthily matched. And here I find an educated man of equal
birth with honorable parts, with fine estates and handsome to behold,
and what is my reward? A puking fool, who answers, "I'll not wed. I
cannot love. I am too young. I pray you pardon me."
Parts here means qualities, attributes, abilities. It is an archaic or rare usage. Today it is retained mostly in the saying
A man of many parts
which the ODO (Oxford Online Dictionary) (link) defines as
A man showing great ability in many different areas.
The ODO also offers the definition of parts as
5 archaic Abilities.
Thus, Honorable parts means 'qualities, attributes, abilities that are worthy of honor'. I assume you know the meaning of 'honorable', since the focus of your question is on parts.
For a more robust definition of parts, we might turn to the OED (Oxford English Dictionary):
15. A personal quality or attribute, especially of an intellectual kind; an ability, gift, or talent.
Usually in the plural.
Now rare, except in man (also woman, lad, etc.) of (many) parts (noun) a man, etc., who is talented or accomplished in many respects.
The 2013 movie Romeo & Juliet which you have referenced retains the phrase honourable parts from the text of Shakespeare's Romeo & Juliet, Act 3, Scene 5. I quote it only to show the original wording:
God's bread! it makes me mad! Day, night, work, play,
Alone, in company, still my care hath been
To have her matched, and having now provided
A gentleman of noble parentage,
Of fair demesnes, youthful, and nobly ligned,
Stuffed, as they say, with honourable parts,
Proportioned as one's thought would wish a man;
And then to have a wretched puling fool,
A whining mammet, in her fortune's tender,
To answer "I'll not wed; I cannot love,
I am too young; I pray you, pardon me."
(see this link, for example, beginning with line 176)
Shakespeare uses a near identical line in Much Ado about Nothing (Act 1, Scene 1), using virtues instead of parts:
A lord to a lord, a man to a man, stuffed with all honorable virtues.
And the OED quotes the same play (Act 5, Scene 2) as saying:
For which of my bad parts didst thou first fall in love with me?
Here, bad parts means bad qualities, or bad character traits, as Shakespeare Navigators puts it (line 60).
Much more evidence can be adduced to show this meaning of parts in Shakespeare's day, but I think that would take this question too much into the realm of Historical English, which the Help Center says is OffTopic. I've quoted a few examples only to show that the meaning you ask about comes straight from the original.