With regard to Abu Bakr (may Allah be pleased with him), there are proven, saheeh hadeeths which state that he is one of those who were given the glad tidings of Paradise. This was previously discussed in the answer to question no. 5852.
With regard to ‘Aa’ishah (may Allah be pleased with her), it is also narrated in saheeh hadeeths that she will be one of the wives of the Prophet (blessings and peace of Allah be upon him) in Paradise.
It was narrated that ‘Aa’ishah (may Allah be pleased with her) said: I said: O Messenger of Allah, who among your wives will be in Paradise? He said: “Verily, you will be one of them.”
Narrated by al-Haakim in al-Mustadrak (4/14). He said: Its isnaad is saheeh although they (al-Bukhaari and Muslim) did not narrate it. al-Dhahabi classed it as saheeh in al-Talkhees. Shaykh al-Albaani (may Allah have mercy on him) said in al-Silsilah al-Saheehah (no. 1142): It is (saheeh) according to the conditions of Muslim. End quote.
‘Ammaar ibn Yaasir (may Allah be pleased with him) said: I know for certain that she is his wife in this world and in the Hereafter.
Narrated by al-Bukhaari (3772).
With regard to Asma’ bint Abi Bakr (may Allah be pleased with her), her virtues are many and great. She bears the noble title of Dhaat al-Nitaaqayn (She of the Two Girdles), as she tore her girdle or waist wrapper in two during the Hijrah (Prophet’s migration to Madeenah) to serve the Messenger of Allah (blessings and peace of Allah be upon him) and his companion Abu Bakr al-Siddeeq, and she was one of the foremost in Islam. Allah, may He be glorified and exalted, said (interpretation of the meaning):
“And those foremost [(in Islamic Faith of Monotheism and in performing righteous deeds) in the life of this world on the very first call for to embrace Islam] will be foremost (in Paradise).
11. These will be the nearest (to Allaah)”
[al-Waaqi ‘ah 56:10-11]
“Not equal among you are those who spent and fought before the conquering (of Makkah, with those among you who did so later). Such are higher in degree than those who spent and fought afterwards”
We have not found any hadeeth which specifically states that she was given the glad tidings of Paradise, except the report narrated from al-Zubayr ibn Bakkaar, who said: She was only called Dhaat al-Nitaaqayn (She of the Two Girdles) because when the Messenger of Allah (blessings and peace of Allah be upon him) and Abu Bakr al-Siddeeq were preparing to set out for Madeenah (the Hijrah), ‘Abd-Allah ibn Abi Bakr came to them in the cave at night with their food supplies for the journey. There was nothing to tie (the bundle) with, so Asma’ tore her girdle in two and tied the bundle with it. The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah will give you in return for that two girdles in Paradise.” So she was called Dhaat al-Nitaaqayn.
This was narrated by Ibn ‘Asaakir in Tareekh Dimashq (6/69) and quoted by Ibn ‘Abd al-Barr in al-Isti‘aab (4/1782) and by Ibn Hajar in al-Isaabah (7/487).
As you can see, this report is mursal and does not have an isnaad. Al-Zubayr ibn Bakkaar died in 256 AH, and he does not say who he heard this hadeeth from.
With regard to the mother of ‘Aa’ishah (may Allah be pleased with her), her kunyah, by which she is best known, was Umm Roomaan. There is a difference of opinion concerning her name. It was said that it was Zaynab or Da‘d. She was the daughter of ‘Aamir ibn ‘Uwaymir ibn ‘Abd Shams, and she was one of the first to come to Islam.
‘Aa’ishah (may Allah be pleased with her) said:
I do not remember my parents ever following anything but the true religion.
Narrated by al-Bukhaari, no. 3905.
With regard to specific testimony that she was one of the people of Paradise, this was narrated in a mursal, da‘eef hadeeth from al-Qaasim ibn Muhammad, who said:
When Umm Roomaan was lowered into her grave, the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever would like to look at one of al-hoor al-‘iyn, let him look at Umm Roomaan.
This was narrated by Ibn Sa‘d in al-Tabaqaat al-Kubra (8/277), by Abu Na‘eem in Ma‘rifat al-Sahaabah (6/3498) and by al-Suhami in Tareekh Jarjaan (1/199). All of them narrated it from Hammaad ibn Salamah, from ‘Ali ibn Zayd, from al-Qaasim ibn Muhammad as a mursal report.
Some students of Hammaad ibn Salamah narrated it as a musnad hadeeth from ‘Aa’ishah, but al-Daaraqutni thought it most likely that the report was mursal.
See: al-‘Ilal al-Waaridah fi’l-Ahaadeeth al-Nabawiyyah by al-Daaraqutni (13/360)
For more on her biography, please see al-Isaabah by al-Haafiz Ibn Hajar (8/206).
And Allah knows best.