是否可以通过ARC2中解析的三元组进行SPARQL查询？

The ARC2 documentation is not really precise. It gives the two following examples:

First, parsing an RDF file:

$parser = ARC2::getRDFParser();
$parser->parse('http://example.com/foaf.ttl');
$triples = $parser->getTriples();

Second, making a SPARQL query on a datastore:

$q = 'SELECT ...';
$rows = $store->query($q, 'rows');

…but would it be possible to make a SPARQL query over the parsed $triples directly?

ARC2 doesn't have an in-memory SPARQL processor, I'm afraid. The SPARQL engine is based on rewriting the query to SQL statements, which is why you need the Store component to run queries.

All you can do to make it a bit more smooth is to skip the parsing step and do a $store->query('LOAD <URL>') to directly import the triples into the store, but you still need the MySQL-backed store, unfortunately.