Moldy Cork??

[NarSki]

My husband & I (new to this world of wine) bought some wine from a local winery and when we opened it up to drink one night we thought the cork looked moldy and it smelled a bit musty. We tasted the wine anyway and it tasted absolutely terrible. So is this why you smell the cork when you open a bottle of wine? Is this a common thing? Unfortunately we gave a few bottles of this particular wine to my parents as a Christmas present. We warned them as they have not opened it yet but I'm wondering it it is likely that those bottles will be bad?