If a person is from Parashara gotra but the girl he loves is of Vasishta gotra, are they siblings? (Since Parashara was the grandson of Vasishta) So the question arises: are the gotras belonging to the same lineage sagotras?
It depends on the pravaras associated with the two gotras as marriage rules are based on pravara associated with gotra eg: mahAbhArata anushasana parva CIV :
A woman of the same Pravaras should also be avoided
Some of the shrautasutras associated with vedas contain a section called "pravaradhyaya" which provides details of pravaras associated with gotras and their marriage rules.
This book contains an excerpt of those sections from shrauta sutras and matsyapurANa. chapter VII of this book has details of gotras having "vashishta" as one of their pravaras and marriage rules associated with them.
Similar details are also mentioned in skandapurANa-dharmaranyakhanDa-ch21 :
8. There is no marriage if there is similarity of Pravaras in five or three sages.(If there is similarity of three sages i.e. if ṃree Pravaras are common, where five sages constitute the Pravaras, no marriage can take place.) If there is similarity of two sages where three sages constitute the Pravaras then also there is a ban. So is the case of Bhṛgu and Aṅgiras. In the remaining cases, even if one sage is common the ban is to be enforced.