Lifted classical planners operate directly on first-order planning tasks to avoid the computationally demanding grounding step. However, lifted planning is typically slower, as planners must repeatedly instantiate ground structures during search. Many core components of lifted classical planning, such as successor generation, axiom evaluation, task grounding, and delete-relaxed heuristics, have previously been studied through the lens of Datalog evaluation. We build upon this line of work and extend it by developing and analyzing an execution model with two levels of parallelism: rule-level parallelism and grounding parallelism. We further specialize this solver for planning-specific workloads with a grounder based on clique enumeration, which we extend to support semi-naive Datalog evaluation. Our experimental evaluation using greedy best-first search with the FF heuristic shows that our implementation already solves more tasks than the baselines on a single core, and the gap widens as additional cores are used. Moreover, on hard-to-ground tasks where on average 97.6% of the total runtime is spent in Datalog execution, the proposed execution model exhibits an average parallel fraction of 92.4%, while achieving up to a 6-fold speedup on 8 cores in practice.