1: "Is it just pivot at the elbow, and rotate down? Like a clock? Should arm be near horizontal at the end?".
Whilst not always possible, a good principle for parrying is to execute the parry with the minimum movement necessary to diminish the strike's effectiveness to an acceptable degree.
An exception to the 'mimimum movement' rule might be in the case where you want to substantially redirect an opponent's arm with the aim of executing a counter. The tradeoff with large-arc parries such as these is that by opening them up in this fashion, you are also opening yourself up to a much greater extent than usual.
If you weren't referring to a chest punch, and you want to parry a strike that's headed to your face, an experienced fighter can render a punch ineffective with a slight rotation of the the glove (often accompanied by head movement to the opposite direction of the fist's rotation). This technique achieves the same advantages as described above, ie: the maintenance of defensive posture and offensive capacity.
For example in an orthodox vs orthodox encounter, a left jab can be redirected by this slight wrist rotation (anti-clockwise if viewed from above), combined with slight head movement to the right. A straight right can be redirected in the same fashion by your left wrist (rotating clockwise if viewed from above). In this case, you may decide to shift your head to the left.
This technique enables you to strike powerfully in return, directly along your opponent's centreline, whilst simultaneously shifting your centreline away from the opponent.
This is of course nowhere as easy to execute as it is to describe, but drilling these movements will pay enormous dividends.
2: "They always say it's a light tap parry, however when the punches come hard, it seems a slight tap won't work during sparring. Sometimes I have to smack/hit it down, to misdirect its fast momentum. Is there any rule of thumb? Better to parry, than to be hit it seems? Honestly I put in high force, since I never know how powerful the punch is when it meets".
The issues you describe here illustrate why it might initially be a good idea to practice catching punches, as a first step towards the greater skill required to parry them.
To catch, don't extend your arms towards the opponent. Maintain your guard and simply rotate your palm out to catch the strike as it arrives. This entails you absorbing more of the strikes energy than a parry, but it is safer in some ways and provides a higher margin for error. As you improve, you can gradually incorporate various parries (inwards, downwards, outwards), which obtain a variety of openings for different counters.
Your opponent's force is to be utilised when parrying. Exaggerated movement (which - rightly or wrongly - is what I read into your term, 'smacking'), is to be avoided, for the reasons already described.
If you keep a close guard, and use your torso to brace your arms, you become far more resilient, stable and safe when catching or parrying. If you extend your arms, you risk having them trapped or slapped down and vulnerable to knockout punches, especially left hooks. Half-extending your arms into a downwards 45 degree 'V' is actually very bio-mechanically weak and experienced boxers will be able to alter the trajectory of their blows very slightly to effectively punch straight past, or even straight through, such a defence. This position also means any strikes of your own will be either very weak, or very slow and loudly telegraphed due to the need to retract prior to punching.
Reaching forwards in this manner to frustrate punches can be effective, but only if you thoroughly appreciate and can cater to the risks involved.
Whilst practising defensive movements alone is certainly a good idea, don't neglect to incorporate the counters enable by these defences into your training. An opponent unconcerned with counters can be far more aggressive and is more likely to overwhelm you. In short, counters become an essential part of defence.
Last point: A very effective and advanced parrying technique is to slip outside (to the right) against an incoming jab, simultaneously parrying it with your left. IF you can achieve this, you have effectively opened up your opponent's side whilst removing their capacity to defend or counter. A key trick here is to get your lead (left) foot immediately to the outside of theirs as you perform the slip.
In summary:
- Maintain defensive integrity via a strong guard and head movement.
- Use minimum movement to catch or parry, to maintain both defensive and offensive capacity.
- Allow the punches to come to you and let your body help to absorb the force of unavoidable impacts.
- Counter simultaneously when possible.
This is a fight between two excellent Muay Thai fighters, one of them the greatest of all time: Buakaw vs Sato 1. It is nonetheless an example of two fighters sustaining repeated heavy head blows largely because of their inability or reluctance to parry or catch, and because of their willingness to stay within range with their arms half-extended.
Acknowledging that Muay Thai fighters and kickboxers have to cover more defensive territory with their hands, this slow-footage examination of Gennadiy Golovkin's technique shows how he tempts his opponent into a poor parry/catch precisely in order to deliver a long hook around their extended fist into the unprotected head (5:40). It also demonstrates how he shifts his head off the centreline whilst countering a jab with his own (4:13), and demonstrates a nice, efficient and compact parry and counter (3:58).
Others on S.E. might have some more specific Muay Thai material, but at first glance, this Muay Thai instructional video provides some good material, and demonstrates the importance of compact parry technique.
