Two men forced their way into the house in Goldie Avenue, Wythenshawe, at about 21:30 BST on Tuesday.
They made the teenager hand over her mobile phone, before attacking the 50-year-old man with an axe.
Police said they were "determined to find the men responsible" and have appealed for witnesses to come forward.
The girl's father sustained head and leg injuries in the attack, and was taken to hospital.
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The pair, who also stole cash, fled in the direction of Styal Road.
Det Insp Lisa Ogden, of Greater Manchester Police, said: "The entire family have been left shaken after this terrifying attack in their own home, and it is utterly despicable that a gun was pointed at an innocent 14-year-old girl.
"We are determined to find the men responsible for this and ensure they are held to account for their actions."
The man carrying the gun was described as white, in his 20s, and speaking with a local accent. The second man was wearing a grey hoodie.
