It's possible the question refers to one of the following, each with its own advantages and disadvantages:
* A specific alternative learning system (ALS): The Philippines has ALS programs designed for out-of-school youth and adults. These might informally be referred to as a "third" option, but they aren't a parallel system to the standard six-year elementary.
* Advantages of ALS: Flexibility, accessibility for those who missed formal schooling, personalized learning pace.
* Disadvantages of ALS: Often lacks resources of formal schools, may not be as rigorous, social stigma associated with not attending formal schooling, potential for lower recognition of qualifications by some employers.
* A specific remedial or enrichment program within a school: Some schools might offer additional support programs for students struggling academically or those needing extra challenges. These could be referred to as a "third" program, but again, this wouldn't be a separate system.
* Advantages of remedial programs: Targeted support for struggling students, improved academic performance.
* Disadvantages of remedial programs: Can stigmatize students, might not address root causes of learning difficulties, added burden for teachers and schools.
* Advantages of enrichment programs: Advanced learning opportunities, stimulating engagement for gifted students, improved overall learning outcomes.
* Disadvantages of enrichment programs: May not be accessible to all students (resource limitations), potential for isolating gifted students from their peers if not managed effectively.
* A misunderstanding or misphrasing: The question itself might be based on a misunderstanding of the Philippine educational system.
To answer accurately, more detail is needed on what specifically is meant by "Third Elementary Education Program." Without that clarification, it's impossible to give a precise analysis of advantages and disadvantages.