Parrying refers to a variety of Yoshi-specific techniques that involve the use of counterattacking against opponents via his unique shield and double jump properties. These techniques vary from game to game, though they are most effective in the original Super Smash Bros. and Super Smash Bros. Melee.
In Smash 64
In Smash 64, Yoshi can deflect any attack by parrying. To perform this, the shield button must be pressed 4 or fewer frames before Yoshi is hit. The window is very small, as the shield button must be pressed for a maximum of 3 frames, otherwise it will fail. The extreme difficulty of performing this technique to avoid attacks consistently makes it rarely seen in competitive play, though players such as tacos and SheerMadness are known for their use of it.
In Melee, Yoshi's parry is performed fully pressing the shield button 5 or fewer frames before Yoshi is hit. In the first 6 frames of shielding Yoshi is invincible, which, upon being hit by an attack, causes the attacker to receive hitlag but not Yoshi. From frames 2-6 he can grab or jump (along with other techniques that follow jumps such as wavedashing or jump canceling), however doing so will disable the invincibility. Frames 3-6 his powershield hurtbox will appear and surrounds a large portion of his body along with him receding further into said hurtbox as the animation progresses, making parrying most reliable when Yoshi's head hurtbox is used and/or very early in the shielding animation; if the powershield hurtbox is hit, Yoshi won't be able to cancel the unshielding animation like other characters.
A method popularized by aMSa to make parrying easier in some cases is to crouch beforehand. This works by lowering Yoshi's hurtbox, making him a smaller target. Once he shields his hurtbox will snap to his shielding animation where he is standing upright. This results in an attack that would otherwise hit a normally standing Yoshi to collide with his now fully invincible hurtbox, triggering a parry.
Yoshi's light shield also has a unique ability. If hit by an attack whilst light shielding, Yoshi will slide away a very far distance (grabbing an edge if he slides offstage). This is a useful technique as it renders the opponent unable to pressure or punish Yoshi, which they would normally be able to do since Yoshi's rolls are slow and short-distanced.
Additionally, Yoshi has a property in which if he powershields one move, he can immediately powershield another move that connects right afterward by continuously light shielding. If this is done against a move that has multiple hits, the shield eventually turns white due to the multiple powershields.
In all games
In Smash 64, Melee, Super Smash Bros. Brawl, Super Smash Bros. 4 and Super Smash Bros. Ultimate, Yoshi can also counterattack opponents using the unique properties of his double jump. Due to the armor properties of his double jump, Yoshi can avoid flinching against aerial attacks and perform any aerial after. This is referred to as a double jump cancel counter. While Yoshi would normally jump away and be unable to punish the opponent in time, the double jump cancel technique allows him to immediately use an aerial against the opponent in Smash 64 and Melee. For this reason, the technique is much more limited in Brawl, Smash 4 and Ultimate, as Yoshi is unable to double jump cancel in those games.