An "environmentally friendly" building with a cafe, gallery and shop is planned for the park near Wakefield.
Peter Murray, executive of the YSP, said it would provide an "important new element" of the park's infrastructure and increase its income.
Arts Council England has given £1.7m to the park and the remainder is to come from fundraising.
A planning application for a second visitor centre has been submitted to Wakefield Council and it is due for completion in late 2017.
The new building has been designed to make the minimum impact on the site, YSP said.
The latest project on the 500-acre estate follows the development of the main visitor centre in 2002, the underground gallery in 2006 and refurbishment of the estate chapel in 2011.
The park attracted more than 400,000 visitors in 2014 the highest number since it opened in 1977, it said.
The development plans were revealed as the park's latest sculpture Wave is being installed.
It is part of the installation Blood Swept Lands and Seas of Red that was seen at the Tower of London in 2014 where 888,246 poppies honoured every death in the British and Colonial forces of World War One.

Derbyshire Wildlife Trust took legal action after Derby City Council approved the track, to be built on part of the Sanctuary nature reserve.
The trust said the development would damage the habitat of rare birds.
An interim order to stop work was issued last week and will remain in place until the judicial review.
Mrs Justice Lang said all equipment had to be moved from the site and the judicial review must take place within the next three months.
Tim Birch, conservation manager at the Derbyshire Wildlife Trust, said: "We are relieved to have got through this latest step in our fight to save the Sanctuary Local Nature Reserve.
"We believe that the development of this local nature reserve sets a very dangerous precedent both locally and nationally which is why we are challenging the decision of Derby City Council to destroy it."
The judicial review will look into the legality of Derby City Council's decision to approve the cycle track.
No-one from the authority was available for comment.
The Sanctuary, which lies close to the Pride Park football stadium, was given the go-ahead by the council in 2003.
Threatened bird species including skylarks and lapwings are visitors to the site, according to conservation managers.

Sherri Papini was still bound with restraints when she managed to flag down a car on Thursday some 140 miles (225km) from where she was last seen.
Police, who say she was released by her captors, are looking for two women in connection with her disappearance.
Ms Papini's husband had reported her missing after she failed to pick up their children from day care.
The motive behind her abduction in Redding, northern California, on 2 November, is still unclear.
County Sheriff Tom Bosenko told reporters Ms Papini was receiving medical attention for non-life threatening injuries at a nearby hospital.
"We are very thankful on this day [Thanksgiving] that she was found safe and reunited with her family.''
A $50,000 (Â£40,200) reward is being offered to find those responsible for Ms Papini's capture.
Investigators had interviewed hundreds of local residents regarding her disappearance and received some 400 tips.

He retained his position with 41,337 votes, ahead of Conservative Matthew Vickers, who received 25,229.
Those figures were a total of first and second preference votes as there was no overall winner with 50 per cent after the initial round.
The total turnout was 79,829 - 19.73% of the electorate, which was up 5% from the 2012 election.
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More information is available on the Choose my PCC website.
Yorkshire Sculpture Park (YSP) has unveiled plans for a new £3.8m visitor centre at its Bretton Park entrance.
[NEXT_CONCEPT]
Work to build a cycle track in Derby cannot resume until a judicial review is carried out, a judge at the High Court in London has ruled.
[NEXT_CONCEPT]
A missing California mother of two has been found three weeks after she was abducted while out jogging.
[NEXT_CONCEPT]
Labour's Barry Coppinger has been re-elected as Cleveland Police and Crime Commissioner.