The incidents come days after students in Rewari and Gurgaon held protests over a similar demand.
The protest in Palwal saw the parents locking the teachers of the government-run primary school in Khirbi village after alleging that a staff crunch was affecting studies of their wards and that teachers were irregular in taking classes.
Some of them said the children in the village had no option but to travel to Hodal, Bhiduki and Hassanpur for further studies.
District Education Officer Anil Sharma rushed to the scene and assured the protesting parents that there demands will be met.
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"The department has already started the process of recruiting more teachers for the school," he said.
In Jind, the students locked the gate of the Karsola Government High School and launched their protest outside it along with villagers, Julana Police Station in-charge Chandrabahan said.
The principal and teachers were not allowed to enter the school, and several attempts to end the protest were futile, he said.
"Several requests were made to the government to upgrade the school, however, all have fell on deaf ears," he said.
The sarpanch said the gates will not be opened till the government assures us that the school will be upgraded.
"For Class 12, students have to go to schools in Jind and Rohtak, which are far away from the village. This has resulted in girl students dropping out," Rajpal said.
However, Julana education officer Adarsh Rajan said, "A letter has been written to the Education Department for upgrading the Karsola Government High School."
"A school can be upgraded to Class 12, only, if it has at least 150 students and there is no other school within five kilometres. The decision to upgrade this school will be taken by the School Upgrade Directorate," he said.