The short answer, and slightly snarky answer, is that the Bolsheviks did not try to abolish the family in the 1920s. Bolshevik family policy under Lenin did not aim to abolish marriage or family structures, despite some extreme rhetoric on both sides of the equation. The family policy of the 1920s Soviet state was the byproduct of several different intertwined trends, several of which were rather contradictory.
One of the biggest reforms of family policy in the 1920s was the state's assumption of control over the institution of marriage and the legalization of divorce. This was of course a natural byproduct of assuming control over the Romanov state in which marriage and family policy was in control of the Orthodox Church. The Bolsheviks tended to treat secularization as a given, and in this family policy was no exception. The largely male Bolsheviks for the most part did not care much for the details of family policy and often delegated this work to female Bolsheviks like Alexandra Kollontai. A number of these activists expressed their intention to use Bolshevik family policy as a means for a wider policy of achieving equality between the sexes. This would not be a destruction of the family, but rather a thorough reconstruction of it along Marxist lines. Kollontai would actually argue that this new marriage would be:
a new form of relationship between men and women at comradely and heartfelt union of two free, independent, wage-earning, equal members of communist society.
Kollontai would further assert that traditional marriage, understood as of a piece with the old order, was an archaic remnant of a system of exploitation in the home.
Both her contemporaries and even in some quarters today have taken the statements of Kollontai on the nature of marriage and relations between the sexes as evidence that early Bolshevik regime sought to fundamentally recast gender roles. However, the opinions of these female activists carried relatively little weight within the corridors of power of the emerging Soviet state. The special female sections of the Communist Party, the Zhenotdel set up in 1919 were often the source of mockery by their male colleagues and enjoyed relatively few resources from the state. In a pattern soon to become familiar to other states' legal guarantees of gender equality, rhetoric seldom lived up to reality; marriage might be a secular institution, but Soviet society remained quite patriarchal. Activists, whether in private or in the Zhenotdel, could earn the wrath of the state if they pushed for women's issues too harshly, often earning the charge of "practicing separatism," a claim that carried particular resonance for a generation accustomed to the factionalism of the RSDLP at the turn of the century. Many of the male Bolsheviks' championship of equal rights and sexual freedoms was relatively thin and very much of its time. For example, when the German KPD activist Klara Zatkin appealed to Lenin for permission to organize Hamburg's female prostitutes, Lenin responded with a scathing critique. Lenin argued that prior attention the KPD paid to prostitution was a tactical decision to underscore the moral hypocrisies of bourgeois society. The Soviet leader then shot down any connection between Communism and increased sexual liberalization:
It seems to me that these flourishing sexual theories, which are mainly hypothetical, and often quite arbitrary hypotheses, arise from the personal need to justify personal abnormality or hypertrophy in sexual life before bourgeois morality, and to entreat its patience. This masked respect for bourgeois morality seems to me just as repulsive as poking about in sexual matters. However wild and revolutionary this behavior may be, it is really quite bourgeois. It is, mainly, a hobby of the intellectuals and of the sections nearest to them. There is no place for it in the party, in the class conscious, fighting proletariat.
Lenin wrote similarly sharp rebukes to Kollontai over her pamphlets such as "Make way for the Winged Eros," which predicted greater female sexual agency under the auspices of Communism. Rhetorical attacks on the traditional family, such as "Winged Eros"'s call for collective child-rearing shocked a number of male Bolsheviks and many dismissed calls for sexual liberation, or even devoting time to talk about the issues of sex and the family, as an unwelcome bourgeois distraction from building the socialist state. Nor were the activists of the Zhenodtel completely united; many of them deliberately eschewed the word "feminist" as it suggested bourgeois social norms. Within this context, Stalin's conservative turn in family policy, termed the Great Retreat, was not as much of a retreat in terms of practical application and understanding of family policy within the Soviet state. Many of the core tenets of the post-1917 reforms such as secularization of marriage and family policy remained in place under Stalin, suggesting a line of continuity between the Lenin and Stalin era in terms of family policy.
But the anti-family rhetoric of the activists when coupled with the stark dictatorial turn of the Stalin regime made the Lenin era appear to be freer and more transformative than it was in reality. Like the state's decriminalization of anti-sodomy laws, the Bolsheviks' forays into previously religious matters did not necessarily lead to a more liberal society. Activists in family policy found that they had more freedom to act when their male superiors, and the Bolsheviks' core leadership was overwhelmingly male, delegated to them niggling issues like the "Woman Question." As both Zatkin and Kollontai discovered, pushing too far on social and family issues could lead to very sharp rebukes and disciplinary action.
Sources
Healey, Dan. Homosexual desire in Revolutionary Russia: the regulation of sexual and gender dissent. Chicago: University of Chicago Press, 2001.
Hoffmann, David Lloyd. Stalinist Values: The Cultural Norms of Soviet Modernity, 1917-1941. Ithaca, N.Y.: Cornell University Press, 2003.
Wood, Elizabeth A. The Baba and the Comrade: Gender and Politics in Revolutionary Russia. Bloomington: Indiana University Press, 1997.