JOHANNESBURG — South African President Jacob Zuma again survived a no-confidence vote in parliament Tuesday in the most serious attempt yet to unseat him after months of growing anger over alleged corruption and a sinking economy, while his party that has ruled since the end of apartheid continued to fracture. Zuma survived six previous attempts to dislodge him in parliament, but this was the first held by secret ballot after parliamentary speaker Baleka Mbete made the surprise decision to allow it. Opposition parties hoped it would encourage legislators with the ruling African National Congress party to vote, without fear of retaliation, against Zuma, under whose leadership the economy has slipped into recession. Zuma’s ties to the Gupta family, immigrant businessmen accused of trying to manipulate government leaders and state companies for financial gain, also have stirred public anger.