The motion was narrowly quashed by 97 votes to 90 with 5 abstentions.
The vote is a blow to those who have long campaigned in favour of statutory paternity leave in Switzerland.
In a statement released to news agencies including ATS, the lobby group ‘Young parents for a modern family policy' said: “The denial of reality by parliament is an affront to young parents of our country and to equality between men and women.
“We are not ready to continue to tolerate the proven suspicion parliament has for the interests of our generation,” it said.
Switzerland currently has no statutory paternity leave, though fathers are allowed one or two ‘family days' for the birth of their child.
Over the past few years several politicians and lobby groups have tried to put the issue on the political table, but to little effect.
Last year the group Travail.Suisse called for a minimum of 20 days paternity leave to be paid at 80 percent of normal pay.
And just last week an advisory body on women's rights published a report calling for the government to introduce 24 weeks shared parental leave, with some of it reserved by law for fathers.
Left-wing politicians have long called for a change to the law, but the Swiss political right sees no need to legislate on the subject.
According to PLR politician Regine Sauter, many companies in Switzerland already offer some form of paternity leave and it's not the government's place “to intervene”, reported ATS.
Switzerland lags behind other European countries on the issue.
Paternity leave across the EU averages at 12.5 days, though some countries such as Sweden allow far more through shared parental leave policies.
Mothers in Switzerland are entitled to 14 weeks paid maternity leave.