What you have done here is mixed up two constructions. The 'so' form goes like this:
'My parents are art lovers, and so am I'
Note that the 'so' form requires an inversion of the subject and verb : 'so am I'.
Alternatively, you can use the 'too' construction, especially if you use two sentences:
'My parents are art lovers. I am too'
'My parents are art lovers and I'm one too.'
In the second alternative, you can use the pronoun 'one' to represent the phrase 'an art lover' and avoid the repetition.
Note that 'too' comes at the end of the phrase, so there is no inversion: 'I am too'.