I could not find anything that said yes or no
Women were permitted to serve, but by all accounts the portion of women serving during the Cold War was very low. Most estimates I have seen put it at between 10,000 and 20,000 depending on the year, with a slight increase during the 1980s. A Defense Intelligence Agency Report from 1976 says there were as few as 659 women in the SA in 1959, out of about 4 million standing strength.^(1) Frustratingly, all of the estimates I have found so far are Western intelligence estimates and not Russian primary sources. Western assessments of the Soviet military at the time were rather fraught and sometimes missed important context, or were entirely incorrect.
What we do know for certain from the documentary record is as follows.
The laws governing recruitment for the Soviet military did permit women in the service. Further, notionally the law after 1967 did permit the conscription of women in wartime. In detail: Until 1967 service was governed chiefly by the 1939 Law on Universal Military Service (as well as the amendments of 1940, 1941, 1943, 1950, and 1954). The 1939 law permitted the mobilization of "women having medical, veterinarian, and special technical training" but unless I am mistaken it did not permit their forcible conscription. Certainly during WW2 the Soviets undertook concerted recruitment and mobilization efforts for these women, but not conscription. As I will address further on in this post, the evidence is overwhelming that the Soviet government and society as a whole generally this as an undesirable anomaly, and went to great lengths to return to "business as normal" after the end of the war.
Article 16 of the 1967 Law tells us that
Women 19 to 40 years of age who have medical or specialized training can be taken into military service in peacetime, recruited for refresher training periods or admitted as volunteers for active duty.
In wartime women can be drafted into the USSR Armed Forces by decision of the USSR Council of Ministers to perform auxiliary or specialized service.
Women were also not permitted to enter the officer academies, which severely curtailed their entry into most career fields. The aforementioned DIA report claims that women were also proscribed from most armed positions and positions on combat ships.^(2) This tracks with some firmer evidence we have from the Soviet involvement in Afghanistan in particular (the subject of one of my previous answers) suggesting that virtually all of the women who served in OKSVA were either medical staff or admin staff. This previous answer, citing Mark Galeotti's social history of the Soviets in Afghanistan, estimates that during that war about 1.5% of the SA was female. Also according to this DIA report, what little active recruitment there was for women in the post-1967 SA was focused on unmarried & childless 19-25 year olds.^(3)
One curious piece of evidence, which I am lumping in with "the laws" even though it was really a Soviet Army regulation, does hint at some acknowledgement of female servicemembers in theory. The 1975 edition of the Устав Внутренней cлужбы Вооруженных cил СССР (Internal Service Charter of the Armed Forces of the USSR) describes a range of daily responsibilities, duties, and protocols for the armed forces irrespective of specific billets or combat taskings. This includes protocols for rewarding soldiers for good performance, as well as for non-judicial punishments for crimes. There are a handful of curious gender variances in these two sections. Men are permitted to be rewarded with either short blocks of leave or a promotion to the next appropriate rank within their billet, among other rewards; women have neither of these reward enumerated in their section. Also, female officers are not permitted to be put in the jailhouse/stockades while awaiting judicial review for crimes (a privilege also reserved for ship captains).^(4)
Volumes have been written about gender norms in Soviet Russia and the Union in general, such that trying to summarize the entire social construction(s) & interpretations of gender in the USSR is vastly beyond the scope of this post.^(5) Speaking in rough generalities, we can say that the (often state-backed) hegemonic conception of gender roles in the USSR was fairly conservative, despite the genuine egalitarian bent in educational and workforce policies. This patriarchal tendency was not static with time of course - the Leninist, early Stalinist, and Khruschev eras are particularly considered to be strong reformist periods.
However, the Great Patriotic War period informed all aspects of future Soviet military policies much more strongly than any other period in Soviet history. The war period was also marked by a staunchly conservative turn in gender roles - despite the massive participation of women in the Red Army during the war! As a number of authors including Reina Pennington, Roger Markwick & Euridice Charon Cardona have argued, Soviet policy towards fighting-age women in the war was sort of pulling in two directions (with one direction or another dominating as the exact wartime needs shifted).^(6,7) On the one hand, the realities of the manpower collapse in 1941 necessitated expanding the recruitment pool. On the other hand, war propaganda centered on shoring up patriotic sentiment through references to a specifically Russian past, and this included more traditional gender norms like motherhood and a vision of women "holding down the fort" while the men served at the front. These two were never entirely at odds - there was a strand of propaganda which emphasized the motherly, nurturing aspects and the patriotic duty exhibited by female nurses.
Even before the end of the war Soviet authorities started to confront a demographic crisis caused by the colossal death toll of the war, propaganda turned sharply back towards motherhood. As demobilization began, virtually every woman in the Red Army was demobilized by the end of 1946. A number of laws and policies to incentivize childrearing "locked in" a more maternal role for Soviet women, although they were also heavily encouraged to stay in the workforce to help make up for the loss of working-aged men during the war. Khruschev's reforms did attempt to bring more women into political engagement, particularly bodies like the Komsomol, but I am not aware of any evidence that this extended to military service.^(8) Several of the higher-profile female aviators recall being pushed out of the service during the mid-late 1950s, during Khruschev's tenure as First Secretary.^(9)
Where is this going? Well, certainly in the 1960s time period you're asking about, we have some strong cultural evidence suggesting that even though it would have been lawful for women to volunteer and even be recruited for non-combat positions, in actuality they were strongly dissuaded from such.
Conclusion: In the absence of firm, primary-source numbers on the number of women in the Armed Forces of the USSR in the 1960s we can estimate from secondary sources that there were likely no more than 10,000 serving, and probably far fewer before 1967.
hey thenks