In this question it is said that the three strands of the poonal signify the debts to parents, teachers and sages.
If so, aren't these debts also meant for women?
Why is it that women do not wear Poonal?
The Yajyopavita is worn by those who after Upanayana study Vedas and live life by practicing Vedic duties.
In the olden days, many women underwent Upanayana and studied Vedas. They were called Brahmavadinis. But, many women who had no interest in learning Vedas, underwent Marriage.
Even older smritis like Harita Smriti mentions women having Upanayanam. See this link- http://www.hinduismtoday.com/modules/smartsection/item.php?itemid=3852
Slowly, as women did not study Vedas and Vedic chanting. They did not undergo separate Upanayana. Instead, marriage itself acted as Upanayana, which gave them right to perform Vedic sacrifices by sitting with their husband and assisting him.
Manu (2.67) also says that, marriage itself is a "Samskara" and husband is like a Guru and cooking food itself is like a fire-ritual. That is, Upanayanam is not compulsory for women. It depends on their individual temperaments.
In addition to the above answers, I would like to add mine:
In Vedic Era, which continued upto Mahabharata period, i.e, 3102 B.C., women used to perform SANDHYA rites. Thereafter, due to reasons not available, women were prohibited from doing so.
In Srimad Ramayana, we come across an indirect reference to Sita performing Sandya rites.
While searching for Sita in Lanka, Sri Hanuma enters Ashoka Garden, where observes the beautiful ambiance of the Garden.
Sri Hanuma sees a mountain, which was surrounded in all directions by peaks, spread by caves and covered by different trees and a river, which fell down from the mountain.
He thought that Seetha would definitely come to that garden if she were to be alive. He also thought that Seetha , interested in the rites of Sandhya time, will definitely come to that river, with the auspicious water, for Sandhya rite.
संध्या काल मनाः श्यामा ध्रुवम् एष्यति जानकी | नदीम् च इमाम् शिव जलाम् संध्या अर्थे वर वर्णिनी ||
(Sundara Kanda 14th Sarga 49th Sloka)
"The ever youthful one with the best complexion, Seetha interested in the rites of Sandhya time will definitely come to this river with the auspicious water for Sandhya rite."
It can be worn even today. Some institutions like Arya samaj and Gayatri parivar are doing some work to promote it.
As per Haarita sutra,
those women who are Brahma Vaadini have right for initiation, Vedic studies , Fire establishment and bhiksha etc. द्विऽविधाः स्त्रियो ब्रह्मवादिन्यः सदीवध्वश्च । तत्र ब्रह्मवादिनीनामुपनयनम्योन्धनं वेदाध्ययनं स्वगृहे भिक्शाचर्या चेति ( 21:23)
As per Yama Smriti, पुरा कल्पे कुमारिणां मोञ्जोबन्धन मिष्यते । अध्यापनन्श्च वेदानां सावित्रीवाचनन्तथा ॥ पिता पितृव्यो भ्राता वा नैनामध्यापयेत् परः | In olden times tying of girdle was prescribed for girls. So also study of vedas and uttering of Gayatri ( Savitur) preferably from father , brother or uncle. She is allowed for begging alms within her house limits.
Yama refers to existence not only in olden days but during his times as well by his sentence पुराकल्पे ( यथैष्यत) तथा (अधुनाऽपि) इष्यते. The word तथा is specially put by Yama by way of comparison between old time and his own time so he says that during his time girls should learn from father, brother etc and she dont need to wear deer skin, bark etc. If Yama had intended to say that girls should not be initiated in his time unlike olden times he would say something like इह कल्पे तासामेवं न कुर्यान्मतिमान् क्कचित् ॥ As he uses verbs in present tense so we can make out that he prescribed rules for his own days and was the case in olden days too.
**Gobhila says that Bride should wear the sacred thread when the bridegroom leads her to the altar. "प्रावृतां यज्ञोपवीतिनीमम्युदानयन् जपेत् सीमोऽदाद् गन्धर्वायेति " ( Gobhila Grhya sutra 2:1:19)
यज्ञोपवीतिनाऽचान्तोदकेन कृत्यम् ( Gobhila Grhya sutra 1:1:2)
Like the women are eligible for vedas, they are eligible for brahmavidya too. (यथाधिकरः श्रोतेषु योषितां कर्म सुश्रुतःएवमेवानुमन्यम्ब ब्रह्माणि ब्रह्मवादिताम् Yama Smriti )
Brahamchari is one who is engaged in the study of vedas. ब्रह्मचर्येण ब्रह्म वेदः तदध्ययनार्थमाचर्यम् Atharva veda sAyana bhashyam 117.17**
A brahmacharin should be married to a brahmacharini. ब्रह्मचरिणां च ब्रहचारिणीभिः सह विवाह् प्रशस्यो भवति - Pranavavada of maharshi Garga The Rig-Veda (V, 7, 9) refers to young maidens completing their education as brahmacharinis and then gaining husbands… Rv. iii (55, 16) mentions unmarried learned and young daughters who should be married to learned bridegrooms.