Firstly: 

If jumping out of the aeroplane with a parachute is done with the aim of training in modern methods of fighting and warfare, there is nothing wrong with it; rather it is something that is enjoined, because Allah says: 

“And make ready against them all you can of power, including steeds of war (tanks, planes, missiles, artillery) to threaten the enemy of Allaah and your enemy…”

[al-Anfaal 8:60].

Shaykh al-Sa‘di (may Allah have mercy on him) said: 

This means, “make ready” against your kaafir enemies who are trying to destroy you and eradicate your religion “all you can of power”, meaning everything that you are able to prepare of mental and physical strength, and all kinds of weapons and the like which will help you to fight them. That includes different kinds of industries in which various kinds of weapons and equipment are made, such as guns, machine guns, bullets, aircraft, land vehicles, boats and ships, fortifications, citadels, trenches, defensive equipment, and everything by means of which the Muslims may advance and defend themselves against the evil of their enemies; learning how to shoot; courage and training. 

Hence the Prophet (blessings and peace of Allah be upon him) said: “Verily, strength is archery.” That includes preparing the transportation means that are needed at the time of fighting. Hence Allah says (interpretation of the meaning): “including steeds of war (tanks, planes, missiles, artillery) to threaten the enemy of Allaah and your enemy…”[al-Anfaal 8:60]. This is the reason which is present in our times, which is to threaten the enemy and the ruling depends on the reason. 

If there is something that will cause a greater deterrent than that, such as land and air craft which are prepared for fighting which could give the Muslims an extra edge against their enemies, then it is enjoined to prepare them and strive to obtain them, and if they cannot be acquired except by learning how to manufacture them, then it becomes obligatory to do that, because whatever is essential to enable one to carry out an obligation is also obligatory. End quote. 

Tafseer al-Sa‘di, p. 324-325 

But if jumping out of the aircraft is done merely for fun and excitement, then it is not permissible. The least that can be said about it is that it is makrooh in the case where it is thought most likely to be safe. If it is thought most likely that the person will die or be harmed, physically or mentally, then it is haraam in that case. 

It was narrated that Abu ‘Imraan al-Jawni said: One of the companions of Muhammad told me, when we were going out on a campaign towards Persia: The Messenger of Allah (blessings and peace of Allah be upon him) said: “If a person spends the night on a roof which has no surrounding wall, and falls down and dies, no one is responsible for his death and if a person travels by sea when it is stormy and dies, no one is responsible for his death.”

Narrated by Imam Ahmad in his Musnad, 20224; classed as hasan by al-Albaani. 

Ibn Battaal (may Allah have mercy on him) said:

What it means, in sha Allah, is that he is no longer under the protection of sharee‘ah, because he brought his own destruction upon himself and he exposed himself to danger, and did not protected himself, so he is not protected by sharee‘ah, and no one could be regarded as not protected by sharee‘ah except in case of kufr. End quote. 

Sharh al-Bukhaari, 5/89 

Even though the hadeeth definitely does not mean that the person who does that is a kaafir, it is one of the hadeeths which serves as a warning to those who commit major sins; it does not refer to that kufr which puts one beyond the pale of Islam. 

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: 

It is permissible when there is no risk. This is the well-known scholarly view. And if it is thought that it is most likely safe, then land and sea are regarded as the same. End quote. Fath al-Baari, 6/88 

Shaykh Ibn ‘Uthaymeen was asked: 

Is it permissible to take risks or put oneself in danger, as we see nowadays in some kinds of extreme sports which may lead to the death of those who engage in them? 

He replied: 

This is haraam. It is not permissible for a person to risk his life in doing things in which there is the fear of death or harm, because Allah says (interpretation of the meaning): “And do not kill yourselves (nor kill one another). Surely, Allaah is Most Merciful to you” [al-Nisa’ 4:29]. If Allah has forbidden that and said “And do not kill yourselves”, then anything that leads to death or leads to harm is also haraam. The Prophet (blessings and peace of Allah be upon him) said: “Your blood, your wealth and your honour are sacred to one another.” Just as it is not permissible for a person to transgress against others, it is not permissible for him to transgress against himself by exposing himself to that which may cause death or harm. End quote. 

Fataawa Noor ‘ala al-Darb, 13/252 

To sum up: 

The person who does this kind of action or takes this kind of risk, if it is for the purpose of training in modern methods of fighting in order to prepare strength against the enemy, there is nothing wrong with him doing that; rather it is enjoined in sha Allah, but the one who is in charge of that should choose those who are suited for this kind of mission and are physically and mentally prepared for it and able to do this kind of mission and train them properly before letting them embark on this kind of mission for real. 

If he dies or is harmed by doing that, he is not killing himself; rather he will be rewarded for his intention and his action, in sha Allah. 

As for doing it as a sport and for fun, as is common nowadays, it is not permissible. 

And Allah knows best.