Ruling on kissing one’s wife in the street in front of people
Please note that you will never be able to live your life without sins and mistakes. This is the nature of man. The Prophet (peace and blessings of Allaah be upon him) said: “Every son of Adam commits sin, and the best of those who commit sin are those who repent.” Narrated by al-Tirmidhi, 2499; Ibn Maajah, 4251; classed as hasan by al-Albaani in Saheeh al-Tirmidhi. This hadeeth clearly indicates that man will inevitably commit sin, but what matters after that is what he does about his sin. The believer repents to Allaah from all his sins, gives them up and asks Allaah for forgiveness every time he commits a sin, and he regrets doing it and resolves sincerely not to do it again. If you do that, then understand that Allaah is Oft-Forgiving, Most Merciful; He forgives all sins for the righteous believer who repents sincerely and admits his sin humbly before Allaah. Allaah says (interpretation of the meaning):
“Say: O ‘Ibaadi (My slaves) who have transgressed against themselves (by committing evil deeds and sins)! Despair not of the Mercy of Allaah, verily, Allaah forgives all sins. Truly, He is Oft‑Forgiving, Most Merciful”
The sin that you have admitted to is not zina with your wife! Rather it is kissing her in front of other people, because there is no zina with one's wife, rather zina is done with a woman whom it is not permissible for a man to touch. But in the case of a wife, it is permissible to touch her.
It is not permissible for a man or a woman to speak to anyone else of what happens between them in bed which no one else should see but them, because of the evils and temptations that result from that and because it opens the door to the shaytaan. This has to do with one who speaks about what he did with his wife, so how about one who does things in front of people where they can see him!
Shaykh Muhammad ibn Ibraaheem (may Allaah have mercy on him) said in his Fataawa (10/277) that kissing one’s wife in front of people is not permissible.
As for the expiation for this sin, there is no expiation except sincere repentance and firm resolve not to do it again, and true regret for having committed this sin.
As for confessing to your parents, there is no need for that. The sin that you have admitted to is a sin against Allaah, so confession should be made to Allaah. It is between you and Allaah and you should not tell anyone about it, but be sincere in your repentance to Allaah and Allaah will forgive you. Allaah is Oft-Forgiving, Most Merciful.
The fact that your wife is the one who told you to do this deed does not mean that she is not a righteous woman or that she does not fear Allaah. You also agreed with her to do it, and you did not stop even when that man saw you. So you should also shoulder your share of responsibility for what you did.
May Allaah help you to do that which He loves and which pleases Him.
And Allaah knows best.