The support staff are protesting against the county council's proposal to only pay them during term time.
Unions ATL and Unison say the move would see some people lose about 20% of their salary.
The council argues it could face equal pay claims from other employees if it does not make the cuts.
Reaction to the teaching assistant strikes and other news from north-east England
Picket lines were operating outside about 50 schools in the area, according to union officials, with lobbying of a council meeting planned for Wednesday.
Jen Linford said she felt "so under-valued" that she quit her teaching assistant role after 37 years.
Joining a picket line in Blackhall, near Peterlee, Mrs Linford said: "If the council walked in our shoes they would realise we deserve more not less. I think it's shameful.
"I still have bills to pay, but morally I just thought that I deserved more and I'm not going to do this anymore. So currently, I work in Asda.
"I have had cuddles this morning from children that I left in July. It's very upsetting.
"I feel the skills that I gained over many years are not being used as they should."
Unison General Secretary Dave Prentis said: "The way these teaching assistants have been treated is nothing short of disgraceful.
"The number of schools closing as a result shows how strongly staff feel.
"They regret any disruption to parents but feel they have no other option."
Councillors voted in May to dismiss the classroom assistants and re-employ them on term-time contracts from January 2017, with one year's compensation.
Following discussions, that offer was increased to two years' compensation with the new terms to be introduced from April 2017.
Members of the GMB and Unite unions voted to accept it.
John Hewitt, the county council's corporate director for resources, said: "The legal advice is unequivocal - the status quo is not an option.
"There is a real and substantial risk from costly equal pay claims which are already materialising and which use teaching assistants as the comparator - most people are not paid for hours and weeks they don't work - so addressing this is also an issue of fairness."
On Monday the council said 37 of the county's 243 schools - mainly nurseries and special schools - would be closed with 76 seeing classes interrupted.
More than 100 schools are either closed or cutting classes as a 48-hour strike by teaching assistants in County Durham gets under way.