Is it permissible to rest on the knees and toes when sitting for the tashahhud and between the two prostrations?

Firstly:

Kneeling or resting on the knees is one specific way of sitting, because the one who does that is in a state of alertness and in a hurry. Mostly that happens when one is arguing, or in a state of fear and suspense, and the like.

Az-Zajjaj (may Allah have mercy on him) said:

What is meant by kneeling or being on one’s knees is resting on one’s knees and toes. Ma‘ani al-Quran wa I‘rabuhu by az-Zajjaaj (4/435).

Imam Ibn Sidah, in al-Mu‘jam al-‘Azim, wrote a section explaining different ways of sitting, in which he said: Kneeling means sitting resting on one’s knees when arguing and the like. End quote from al-Mukhassas (3/333); Lisan al-‘Arab (14/131).

Ash-Shurayshi (may Allah have mercy on him) said: Kneeling means sitting resting on the knees. Sharh Maqamat al-Hariri (2/371).

To sum up:

Kneeling is a specific way of sitting, but it is not usually resorted to in a calm situation; rather it is something that is rarely resorted to, or is resorted to when needed.

Secondly:

The Sunnah way of sitting between the two prostrations is iftirash (sitting muftarishan, which means sitting with the right foot held upright on the toes, whilst sitting on the left foot), and this is also done in the tashahhud of a two-rak‘ah prayer; and tawarruk (sitting mutawarrikan, which means slipping the left foot under the right calf, with one’s buttocks on the ground) in the final tashahhud of a three- or four-rak‘ah prayer, as has been explained in the answer to question no. 13340 .

But if the worshipper cannot sit muftarishan, or that is extraordinarily difficult for him, for any reason, then he may sit in whatever way is easy for him, because Allah, may He be exalted, says (interpretation of the meaning): {So fear Allah as much as you are able } [at-Taghabun 64:16], and because the Prophet (blessings and peace of Allah be upon him) said: “If I instruct you to do something, then do as much of it as you are able.” Al-Bukhari (7288).

And he (blessings and peace of Allah be upon him) said to ‘Imran ibn Husayn: “Pray standing; and if you cannot, then sitting; and if you cannot, then lying on your side.” Al-Bukhari (1066).

Thus he explained to him that standing, which is an essential and obligatory part of the prayer, is waived when one is unable to do it, so how about the manner of sitting, which is Sunnah (not obligatory)?

So if a person’s prayer is valid if he does not sit in the Sunnah manner without an excuse, then it is more appropriate that his prayer should still be valid when there is an excuse, as the reward for his intention that he would sit in the Sunnah manner if he were able to will be recorded for him.

Abu Hamid al-Ghazali (may Allah have mercy on him) said: There is no specific way to sit, besides which nothing else could be valid; rather it is valid to sit in whatever way one can.

End quote from Fath al-‘Aziz bi Sharh al-Wajiz (3/494).

Al-Ansari (may Allah have mercy on him) said: Whatever way a person sits in the prayer, it is valid. End quote from Asna al-Matalib (1/164).

Al-Juwayni (may Allah have mercy on him) said: If he rests on his feet and knees, I do not think that there is anything wrong with it. End quote from Nihayat al-Matlab fi Dirayah al-Madhhab (2/215).

Shaykh Mar‘i al-Maqdisi (may Allah have mercy on him) said: Sitting between the two prostrations in any manner is valid, but the Sunnah is to sit muftarishan, resting on the left foot and holding the right foot upright.

End quote from Dalil at-Talib li Nayl al-Matalib (p. 34).

Based on that, if someone is not able to sit in the Sunnah manner between the two prostrations or in the tashahhud, it is permissible for him to sit resting on his knees.

And Allah knows best.