Youth and age are certainly contrasted in Romeo and Juliet, though not necessarily old age. It is rather a contrast between youths and adults.
Youth is portrayed through the young characters of the play, such as Romeo, Juliet, Benvolio, and Mercutio. Juliet is our youngest character, being only 13. We know that Juliet is only 13 because her father points out to Paris that she is not yet fourteen in saying, "She hath not seen the change of fourteen years" (I.ii.9). Likewise, Juliet's Nurse points out that in a little more than two weeks, Juliet will be 14 (I.iii.18-21). We are never told exactly how old Romeo is, but since he and his friends are men roaming around town, we know that he and his friends are at least in their late teens, possibly early twenties. If they were younger, they would still be at home studying under their tutors, which was the way that boys of high society were educated back then.
All of these young characters make extremely emotionally driven, impetuous, and rash decisions. Even Benvolio, who is known for being the peace maker, comes up with the foolish idea to crash the Capulet's ball. We can assume that crashing the ball was his idea because, as the second speaker in the scene, he is the one to lay out the plan in Scene 4 of Act 1. Both Romeo and Juliet also make the rash, emotional decision to marry so suddenly. The whole play is full of youthful, emotionally driven, rash decisions.
In contrast, some of the characters that represent the older, adult generation of the play are Lords and Ladies Capulet and Montague, Friar Laurence, and Prince Escalus. But even these older characters are not necessarily wiser. They, too, are known for making emotionally driven, rash decisions. Lords Capulet and Montague have made the decision to carry on a feud that was actually begun by their ancestors, as we learn from the line, "From ancient grudge break to new mutiny," found in the opening Prologue (3). Their decision to fight actually goes against their own known better judgement, as we learn from Capulet when he declares "and 'tis not hard, I think, / For men so old as we to keep the peace" (I.ii.2-3). Friar Laurence decides against his own better judgement to marry Romeo and Juliet, simply because he hopes it will end the feud. He also makes the irrational decision to fake Juliet's death, rather than to announce her marriage to her father. Even Prince Escalus, who is known as the voice of righteous judgement and pure reason, confesses to making poor decisions. He repents not putting an end to the feud sooner as it cost him Mercutio's life, his own family member.
Hence, we see that while youth is contrasted with the older generation in the play, both the youth and the adults actually make the same foolish, emotionally driven, impetuous, and rash decisions, showing us that so long as you allow yourself to be governed by your emotions, there is actually not a vast difference between youth and age.
Act one. Scene three.
This scene introduces Juliet on stage and explores the theme of youth versus old age and the difference in attitudes between the Nurse, Lady Capulet and Juliet towards love and marriage. The Nurse's uninhibited attitdue towards sex is constrasted with Lady Capulet's reserved discussion of Juliet's proposed marriage to Paris.
The Nurse's bawdy character is also a big contrast to Juliet's naive, youthful spirit as Nurse has a older, courser outlook on life. In Lady Capulet's relationship with Juliet everything seems distant and cold, expecting Juliet's complete obedience in agreeing to marriage. Juliet is clearly reluctant to agree with an arranged marriage but says " It is an honor that I dreamt not of."